Abstract
Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system
We consider the critical mass case \(\int _{{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi \), which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function \(u_0^*\) with mass \(8\pi \) such that for any initial condition \(u_0\) sufficiently close to \(u_0^*\) and mass \(8\pi \), the solution u(x, t) of (\(*\)) is globally defined and blows-up in infinite time. As \(t\rightarrow +\infty \) it has the approximate profile
where \(\lambda (t) \approx \frac{c}{\sqrt{\log t}}\), \(\xi (t)\rightarrow q\) for some \(c>0\) and \(q\in {\mathbb {R}}^2\). This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).
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1 Introduction
This paper deals with the classical Keller–Segel problem in \({\mathbb {R}}^2\),
which is a well-known model for the dynamics of a population density u(x, t) evolving by diffusion with a chemotactic drift. We consider positive solutions which are well defined, unique and smooth up to a maximal time \(0< T \leqq +\infty \). This problem formally preserves mass, in the sense that
An interesting feature of (1.1) is the connection between the second moment of the solution and its mass which is precisely given by
provided that the second moments are finite. If \(M>8\pi \), the negative rate of production of the second moment and the positivity of the solution implies finite blow-up time. If \(M<8\pi \) the solution lives at all times and diffuses to zero with a self similar profile according to [5]. When \(M = 8\pi \) the solution is globally defined in time. If the initial second moment is finite, it is preserved in time, and there is infinite time blow-up for the solution, as was shown in [4].
Globally defined in time solutions of (1.1) are of course its positive finite mass steady states, which consist of the family
We observe that all these steady states have the exact mass \(8\pi \) and infinite second moment
As a consequence, if a solution of (1.1) is attracted by the family \((U_{\lambda ,\xi })\), its mass must be larger than \(8\pi \) and if the initial second moment is finite, then blow-up occurs in a singular limit corresponding to \(\lambda \rightarrow 0_+\).
In the critical mass \(M=8\pi \) case, the infinite-time blow-up in (1.1) when the second moment is finite, takes place in the form of a bubble in the form (1.2) with \(\lambda =\lambda (t)\rightarrow 0\) according to [2, 4]. Formal rates and precise profiles were derived in [8, 12] to be
A radial solution with this rate was built by Ghoul and Masmoudi [26] and its stability within the radial class was established. The framework of the construction in [26] was actually fully nonradial, but for stability a spectral gap inequality only known in the radial case was used. Numerical evidence for this inequality was obtained in [7], and stability for general nonradial perturbation was conjectured in [26]. A related spectral estimate, useful in the analysis of finite time blow-up was found in [15].
In this paper we construct an infinite-time blow-up solution with a different method to that in [26], which in particular leads to a proof of the stability assertion among non-radial functions. The following is our main result:
Theorem 1.1
There exists a nonnegative, radially symmetric function \(u_0^*(x)\) with critical mass \(\int _{{\mathbb {R}}^2} u_0^*(x)\,dx =8\pi \) and finite second moment \(\int _{{\mathbb {R}}^2}|x|^2\, u_0^*(x)\,dx <+\infty \) such that for every \(u_1(x)\) sufficiently close (in suitable sense) to \(u_0^*\) with \(\int _{{\mathbb {R}}^2} u_1\,dx =8\pi \), we have that the solution u(x, t) of system (1.1) with initial condition \(u(x,0)= u_1(x) \) has the form
uniformly on bounded sets of \({\mathbb {R}}^2\), and
for some number \(c>0\) and some \(q\in {\mathbb {R}}^2\).
Sufficiently close for the perturbation \(u_1(x):= u_0^*(x) + \varphi (x)\) in this result is measured in the \(C^1\)-weighted norm for some \(\sigma >1\)
The perturbation \(\varphi \) must have zero mass too.
“Uniformly on bounded sets” of \({\mathbb {R}}^2\) in (1.3) means that for any bounded \(K\subset {\mathbb {R}}^2\)
The expansion of u(x, t) can be made more precise though, and this is explained along the proof of theorem.
The scaling parameter is rather simple to find at main order from the approximate conservation of second moment, see Sect. 2. The center \(\xi (t)\) actually obeys a relatively simple system of nonlocal ODEs.
We devote the rest of this paper to the proof of Theorem 1.1. Our approach borrows elements of constructions in the works [16,17,18, 21] based on the so-called inner-outer gluing scheme, where a system is derived for an inner equation defined near the blow-up point and expressed in the variable of the blowing-up bubble, and an outer problem that sees the whole picture in the original scale. The result of Theorem 1.1 has already been announced in [20] in connection with [16, 18, 21].
There is a vast literature on chemotaxis in biology and in mathematics. The Patlak–Keller–Segel model [35, 44] is used in mathematical biology to describe the motion of mono-cellular organisms, like Dictyostelium Discoideum, which move randomly but experience a drift in presence of a chemo-attractant. Under certain circumstances, these cells are able to emit the chemo-attractant themselves. Through the chemical signal, they coordinate their motion and eventually aggregate. Such a self-organization scenario is at the basis of many models of chemotaxis and is considered as a fundamental mechanism in biology. Of course, the aggregation induced by the drift competes with the noise associated with the random motion so that aggregation occurs only if the chemical signal is strong enough. A classical survey of the mathematical problems in chemotaxis models can be found in [31, 32]. After a proper adimensionalization, it turns out that all coefficients in the Patlak–Keller–Segel model studied in this paper can be taken equal to 1 and that the only free parameter left is the total mass. For further considerations on chemotaxis, we shall refer to [30] for biological models and to [11] for physics backgrounds.
In many situations of interest, cells are moving on a substrate. The two-dimensional case is therefore of special interest in biology, but also turns out to be particularly interesting from the mathematical point of view as well, because of scaling properties, at least in the simplest versions of the Keller–Segel model. Boundary conditions induce various additional difficulties. In the idealized situation of the Euclidean plane \({\mathbb {R}}^2\), it is known since the early work of W. Jäger and S. Luckhaus [33] that solutions globally exist if the mass M is small and blow-up in finite time if M is large. The blow-up in a bounded domain is studied in [1, 33, 39, 40, 46]. The precise threshold for blow-up, \(M=8\pi \), has been determined in [5, 23], with sufficient conditions for global existence if \(M\leqq 8\pi \) in [5] (also see [22] in the radial case). The key estimate is the boundedness of the free energy, which relies on the logarithmic Hardy–Littlewood–Sobolev inequality established in optimal form in [9]. We refer to [3] for a review of related results. If \(M<8\pi \), diffusion dominates: intermediate asymptotic profiles and exact rates of convergence have been determined in [7]. Also see [25, 41]. In the supercritical case \(M>8\pi \), various formal expansions are known for many years, starting with [27, 28, 49] which were later justified in [38, 45], in the radial case, and in [14], in the non-radially symmetric regime. This latter result is based on the analysis of the spectrum of a linearized operator done in [15], based on the earlier work [19], and relies on a scalar product already considered in [45] and similar to the one used in [6, 7] in the subcritical mass regime. An interesting subproduct of the blow-up mechanism in [29, 45] is that the blow-up takes the form of a concentration in the form of a Dirac distribution with mass exactly \(8\pi \) at blow-up time, as was expected from [24, 29], but it is still an open question to decide whether this is, locally in space, the only mechanism of blow-up.
The critical mass case \(M=8\pi \) is more delicate. If the second moment is infinite, there is a variety of behaviors as observed for instance in [36, 37, 43]. For solutions with finite second moment, blow-up is expected to occur as \(t\rightarrow +\infty \): see [34] for grow-up rates in \({\mathbb {R}}^2\), and [48] for the higher-dimensional radial case. The existence in \({\mathbb {R}}^2\) of a global radial solution and first results of large time asymptotics were established in [2] using cumulated mass functions. In [4], the infinite time blow-up was proved without symmetry assumptions using the free energy and an assumption of boundedness of the second moment. Also see [42, 43] for an existence result under weaker assumptions, and further estimates on the solutions. Asymptotic stability of the family of steady states determined by (1.2) under the mass constraint \(M=8\pi \) has been determined in [10]. The blow-up rate \(\lambda (t)\) and the shape of the limiting profile U were identified in formal asymptotic expansions in [12, 13, 47, 49, 50] and also in [8, Chapter 8]. As already mentioned, a radial solution with rate \(\lambda (t)\sim (\log t)^{-1/2}\) was built and its stability within the radial class was established in [26].
2 Formal Derivation of the Behavior of the Parameters
We consider here a first approximation to a solution u(x, t) of (1.1), globally defined in time, such that on bounded sets in x,
for certain functions \(0<\lambda (t) \rightarrow 0\) and \(\xi (t) \rightarrow q\in {\mathbb {R}}^2\), where we recall that
We know that (2.1) can only happen in the critical mass, finite second moment case,
which according to the results in [4, 12, 26] is consistent with a behavior of the form (2.1). Since the second moment of U is infinite, we do not expect the approximation (2.1) be uniform in \({\mathbb {R}}^2\) but sufficiently far, a faster decay in x should take place as we shall see next. We will find an approximate asymptotic expression for the scaling parameter \(\lambda (t)\) that matches with this behavior.
Let us introduce the function \( \Gamma _0: = (-\Delta )^{-1} U. \) We directly compute
and hence \(\Gamma _0\) solves the Liouville equation
Then \(\nabla \Gamma _0(y) \approx -\frac{4y}{|y|^2} \) for all large y, and hence we get, away from \(x=\xi \),
Therefore, defining
and writing in polar coordinates
we find \( {\mathcal {E}}(u) \approx {\partial } _r^2 u + \frac{5}{r} {\partial } _r u \). Hence, assuming that \({\dot{\xi }} (t)\rightarrow 0 \) sufficiently fast, equation (1.1) approximately reads as
which can be idealized as a homogeneous heat equation in \({\mathbb {R}}^6\) for radially symmetric functions. It is therefore reasonable to believe that beyond the self-similar region \(r\gg \sqrt{t}\) the behavior changes into a function of \(r/\sqrt{t}\) with fast decay at \(+\infty \) that yields finiteness of the second moment. To obtain a first global approximation, we simply cut-off the bubble (2.1) beyond the self-similar zone. We introduce a further parameter \(\alpha (t) \) and set
where
with \(\chi _0\) a smooth radial cut-off function such that
We introduce the parameter \(\alpha (t)\) because the total mass of the actual solution should equal \(8\pi \) for all t. However,
as \(t\rightarrow \infty \), where
and \(\chi _0(x) = {\tilde{\chi }}_0(|x|)\). To achieve \( \int _{{\mathbb {R}}^2} {{\bar{u}}}(x,t)\,dx = 8\pi \) we set \(\alpha ={\bar{\alpha }}\) where
Next we will obtain an approximate value of the scaling parameter \(\lambda (t)\) that is consistent with the existence of a solution \(u(x,t)\approx {{\bar{u}}}(x,t)\) where \({{\bar{u}}}\) is the function in (2.3) with \(\alpha ={\bar{\alpha }}\). Let us consider the “error operator”
where \({\mathcal {E}}(u) \) is defined in (2.2). We have the following well-known identities, valid for an arbitrary function \(\omega (x)\) of class \(C^2({\mathbb {R}}^2)\) with finite mass and \(D^2 \omega (x) = O(|x|^{-4-\sigma })\) for large |x|:
and
Let us recall the simple proof of (2.9). Integrating by parts on finite balls with large radii and using the behavior of the boundary terms we get the identities
and then (2.9) follows. The proof of (2.10) is even simpler. For a solution u(x, t) of (1.1) we then get
In particular, if u(x, t) is sufficiently close to \({{\bar{u}}}(x,t)\) and since \(\int _{{\mathbb {R}}^2} {{\bar{u}}}(x,t) dx =8\pi \), we get the approximate validity of the identity
This means
We readily check that for some constant \(\kappa \)
Then we conclude that \(\lambda (t)\) approximately satisfies
and hence we get at main order
We also notice that the center of mass is preserved for a true solution, thanks to (2.10):
Since the center of mass of \({{\bar{u}}}(x,t)\) is exactly \(\xi (t)\) we then get that approximately
3 The Approximations \(u_0\) and \(u_1\)
From now on we to consider the Keller–Segel system starting at a large \(t_0\),
which is equivalent to (1.1). We do this so that some expansions for t large take a simpler form.
In this section we will define a basic approximation to a solution of the Keller–Segel system (3.1). Let us consider parameter functions
that we will later specify. Let us consider the functions
and define the approximate solution \(u_0(x,t)\) as
where \(\chi \) is the cut-off function (5.3). We consider the error operator
where
and next measure the error of approximation \(S(u_0)\).
We have
where
We also have
Let us decompose
For the term \({\mathcal {R}}\) in (3.5) we directly estimate
Then
and thus
For a function \(v(\zeta )\) defined for \(\zeta \in {\mathbb {R}}^2\) consider the operator
The reason for the notation is that for radial functions \( v= v(r)\), \(r=|\zeta |\), we have
which corresponds to Laplace’s operator in \({\mathbb {R}}^6\) on radial functions.
Let \({\tilde{\varphi }}_\lambda (\zeta ,t)\) be the (radial) solution to
given by Duhamel’s formula, where \(E(\zeta ,t)\) is the radial function
and
with \(z = \frac{\zeta }{\sqrt{t}}\), \( y = \frac{\zeta }{\lambda } \).
We then define
The reason to define \(\varphi _\lambda \) for \(t> \frac{t_0}{2}\) is that it gives better properties for the first approximation of \(\lambda \) constructed in Sect. 7. Since \(\lambda (t)\) is defined naturally for \(t>t_0\), we will need to define \(\lambda (t)\) for \(\frac{t_0}{2}<t<t_0\) in an appropriate way (see Proposition 5.1 and Sect. 7). We will write \(\lambda = \lambda _0 + \lambda _1\) where both of these functions are constructed so that they are defined for \(t>\frac{t_0}{2}\). The construction of \(\lambda _0\) is given in Proposition 5.1. In particular \(\lambda _0(t) = \frac{c_0}{\sqrt{\log t}}(1+o(1))\) as \(t\rightarrow \infty \). Note that \(\varphi _\lambda (\cdot ,t_0)\) is not zero.
We define the approximate solution
which depends on the parameter functions \(\alpha (t)\), \(\xi (t)\), \(\lambda (t)\). Correspondingly, we write
We will establish in the next sections that a suitable choice of these functions makes it possible to find an actual solution of (3.1) as a lower order perturbation of \(u_1\).
4 The First Error of Approximation
We will assume the following conditions on \(\lambda \), \(\alpha \), \(\xi \)
where \(\frac{3}{2}<\gamma <2\).
We compute
where
Then
where \({\mathcal {R}}\) is defined in the decomposition (3.5).
Lemma 4.1
Let \(\varphi _\lambda \) be defined by (3.12)-(3.9) with \(\lambda \) satisfying (4.1). Then
We also have
Proof
In terms of the function \({\tilde{\varphi }}_\lambda \) defined in (3.9), with \(r = |x-\xi |\) we claim that
For the proof of this we use barriers. Consider
and note that
for some \(c>0\), \(\delta >0\).
Let \(\chi _{\delta \sqrt{t}}(r,t ) = {\tilde{\chi }}_0(\frac{r}{\delta \sqrt{t}}) \) where \({\tilde{\chi }}_0\in C^\infty ({\mathbb {R}})\) is such that \({\tilde{\chi }}_0(s) = 1\) for \(s \le 1\) and \({\tilde{\chi }}_0(s) = 0\) for \(s \ge 2\). Consider
The function \({{\tilde{E}}}\) (3.11) can be estimated by
where \(h_1(z)\) is a smooth function with compact support. Then E (3.10) has the estimate
where \(h_2(z)\) is a smooth function with compact support.
Then for \(C_1\) sufficiently large
where \(c>0\).
By the comparison principle,
for some uniform constant C. After a suitable scaling, from standard parabolic estimates we also get
With these two inequalities we obtain (4.3).
To prove (4.4) we change variables \(y = \frac{x-\xi }{\lambda }\) in the equation (3.9) and define
We get the equation, after interpreting \(\rho = |y|\), \(y\in {\mathbb {R}}^6\)
where E is defined in (3.10). Differentiating with respect to y and using the bound we already have for \(\nabla _y {\hat{\varphi }}_\lambda \) from (4.4), and using standard parabolic estimates, we get
Using that \(\nabla {\hat{\varphi }}_\lambda (0,t)=0\) we deduce that
which readily gives (4.4). \(\square \)
Lemma 4.2
Assuming (4.1) we have
and
for some \(c\in (0,\frac{1}{4})\).
Proof
Let us analyze the terms involving \(\varphi _\lambda \). We estimate, using Lemma 4.1,
Similarly, by (3.5)
By (4.4)
The other terms in (4.7) are estimated similarly, using the hypotheses on \(\alpha \) and the estimate on \({\mathcal {R}}\) (3.6), and we get
The terms involving \(\psi _\lambda = (-\Delta )^{-1}\varphi _\lambda \) are estimated using the formula
In \( \lambda ^4 S(u_1)\) we have also the term \(- \dot{\alpha }\lambda ^2 U(y) \chi \), which thanks to (4.1) can be estimated as
The remaining terms are estimated similarly, and we obtain (4.5).
The stated inequality (4.6) follows from the Gaussian decay of \(\varphi _\lambda \) in Lemma 4.1. \(\square \)
5 The Inner–Outer Gluing System
Let us consider the initial approximation
built in Sect. 3 for a given choice of the parameter functions \(\lambda (t)\), \(\alpha (t)\), \(\xi (t) \) satisfying (4.1). Here \(u_0\) is the function defined in (3.2) and \(\varphi _\lambda \) that in (3.12). We look for a solution of the Keller–Segel equation (3.1) in the form of a small perturbation of \(u_1\), namely
We write the perturbation \(\Phi \) as a sum of an “inner” contribution, better expressed in the scale of \(u_0\), and a remote effect that takes into consideration the “outer” regime. Precisely, we write
where \(\chi \) is the smooth cut-off
with \(\chi _0\) a smooth radial cut-off function such that \( \chi _0(z) = 1 \) if \(|z|\leqq 1\), \( \chi _0(z) = 1 \) if \( |z|\geqq 2 \). (The same as defined in (2.4).)
Recall S(u) given by
where the operators act on the original variable x unless otherwise indicated. In the computations that follow we will express the equation
for \(\Phi \) given by (5.2), as a parabolic system in its inner and outer contributions \(\phi ^i\) and \(\varphi ^o\). The coupling in that system will be small if \(\phi ^i(y,t)\) decays sufficiently fast in space and time. That can only be achieved for suitable choices of the parameters \(\alpha , \lambda , \xi \) that yield certain solvability conditions satisfied. The set of all these relations is what we call the inner-outer gluing system. Next we formulate this system. It will be necessary to successively refine its original expression by further decomposing \(\phi ^i\) into two contributions with separate space decay, finally arriving at the equations (5.48), (5.49), (5.50) and (5.52) which are the ones we will actually solve.
Let us observe that
where
We use the notation
in the expressions that follow. We expand
We have
and
Recall the notation
and also (3.5)
Then
Therefore
Next we expand
We have
and
Therefore,
Based on the previous formulas we formulate the inner equation
where
We slightly modify the inner equation into the form
where
and
with \(\chi _0\) as in (2.5). Similarly we formulate the outer equation as
where
If \(\phi ^i\), \(\varphi ^o\) is a solution to system (5.5), (5.9), then u given by (5.1), (5.2) satisfies the Keller–Segel system (3.1).
5.1 Choice of \(\lambda _0\) and \(\alpha _0\)
We explain the choice of \(\lambda _0\) in the context of the elliptic equation
where h is radial.
Lemma 5.1
Let h(y) be a radial function such that
for some \(\gamma >4\) and satisfying
Then there exists a radial solution \(\phi (y) \) of equation (5.11) such that
and
Proof
Defining \(g = \frac{\phi }{U}- (-\Delta )^{-1} \phi \) we obtain the equation
Assuming \(\gamma >6\) we choose the radial function g defined by
and using (5.12) we get
Now we solve Liouville’s equation
Multiplying (5.17) by \(|y|^2\) and using (5.13) we see that
with \(Z_0\) defined in (3.4). Then by the variations of parameter formula we find that (5.18) has a unique solution \(\psi \), which satisfies
Then we see that \(\phi \) defined by \(\phi = U g + U \psi \) satisfies (5.11), (5.14) and (5.16) because \(\phi = - \Delta \psi \) and \(\psi \) has the decay (5.19).
If \(4<\gamma \le 6\) we do almost the same, except that we define
\(\square \)
Remark 5.1
We observe that \(L[Z_0]=0\). This can also be seen in the context of the Lemma 5.1, where \(\phi =Z_0\) which corresponds to g being constant. Indeed, suppose \(g \equiv 1\). Then from (5.18) \(\psi = -1 - \frac{1}{2}z_0\), where \(z_0\) is defined in (9.2). This gives \(\phi = U g + U \psi = -\frac{1}{U}z_0 = -\frac{1}{2}Z_0\). This shows that \(L[Z_0]=0\).
If h doesn’t satisfy the zero second moment condition (5.13), then a solution still exists but with worse decay and non-zero mass. More precisely, if h is radial, \(\Vert (1+|y|)^{\gamma } h(y) \Vert _{L^\infty ({\mathbb {R}}^2)} < \infty \) for some \(\gamma > 6\), and satisfies only (5.12), then one can construct a solution \(\phi \) to (5.11), but any such solution has the estimate
so worse decay than the one in (5.14). Moreover, the mass of \(\phi \) becomes
For the inner equation (5.5) it is then natural to impose that the first error \(S(u_1) \chi \) satisfies the second moment condition
The next lemma gives a way of expressing the second moment of \(u_1\).
Lemma 5.2
Let \(u_1\) be defined in (3.13). Then
where E, \({{\tilde{E}}}\) are defined in (3.10), (3.11).
Proof of Lemma 5.2
Using (2.11) we see that
But recall that \(\varphi _\lambda (x,t) = {\tilde{\varphi }}_\lambda (x-\xi (t),t)\) where \({\tilde{\varphi }}_\lambda \) satisfies (3.9). Multiplying that equation by \(|\zeta |^2\) and integrating on \({\mathbb {R}}^2\) results in
Therefore
and then
But from the formula for \( \partial _t u_0\) (3.3) and the definitions of E and \({{\tilde{E}}}\) (3.10), (3.11) we get
Hence
Replacing this in (5.21) we obtain (5.20). \(\square \)
In the definition (3.13) of \(u_1\) we will stress the dependence on the parameters by writing \({\textbf{p}} = (\lambda ,\alpha ,\xi )\) and \(u_1 = u_1({\textbf{p}})\). At this point we would like to construct \(\lambda _0\) and \(\alpha _0\) so that setting \({\textbf{p}}_0 = ( \lambda _0, \alpha _0, 0 ) \) we have
for some \(\sigma >0\). The reason for allowing in (5.23) an error is that it is difficult to solve with right hand side equal to 0 and a remainder of size \(O(t^{-\frac{3}{2}-\sigma })\) with \(\sigma >0\) is sufficiently small to proceed with the rest of the construction.
Assuming that (5.22) holds, we get
It turns out that the main terms in the expression for \(\int _{{\mathbb {R}}^2} S( u_1 ) |x-\xi |^2 dx\) are the first two. So the equation
is at main order given by
It will be shown later that
see Lemma 7.5, where \(\Upsilon \) is given in (2.7), so that the equation we want to solve becomes at main order,
In §7 we will show that
see Corollary 7.1. Using (5.25) we see that
so that the equation for \(\lambda \) is at main order
One can check that \(\lambda ^*(t) = \frac{c_0}{\sqrt{\log t}}\), where \(c_0>0\) is an arbitrary constant, is an approximate solution. Indeed
The error left out in the approximation (5.26) is too big. We give next a result that shows that for an appropriate modification of \(\lambda ^*\) we can achieve a smaller error. Let us write \({{\tilde{E}}}(\lambda )\) the expression defined in (3.11) with the explicit dependence on \(\lambda \).
Proposition 5.1
Let \(c_0>0\) be fixed. For \(t_0>0\) sufficiently large there exists \(\lambda _0:[\frac{t_0}{2},\infty ) \rightarrow (0,\infty ) \) such that
for some \(\sigma >0\). Moreover, for arbitrarily \(\varepsilon >0\) small, \(\lambda _0\) has the expansion
as \( t \rightarrow \infty \).
We will prove this result in §7.1.
Once \(\lambda _0\) is constructed in Proposition 5.1 we choose \(\alpha _0\) so that (5.22) holds, by imposing
We note that by (2.6), (5.27) and (5.24) we get
as \(t\rightarrow \infty \). A byproduct of the proof of Proposition 5.1 is that
and from this and (5.28) we get
As a corollary of Proposition 5.1 we get:
Corollary 5.1
Let \({{\textbf {p}}}_0=(\lambda _0,\alpha _0,0)\) with \(\alpha _0\) defined by (5.22) and \(\lambda _0\) be given by Proposition 5.1. Then
for some \(\sigma >0\).
Proof
Using Lemma 5.2 we have
for some \(\sigma >0\), since \({\dot{\alpha }}_0(t) = O( \frac{1}{t^2 \log t})\) and
by (5.24) and a direct estimate for the remaining terms in E (c.f. (3.10)). \(\square \)
5.2 A further improvement of the approximation
We introduce a correction \(\phi _0^i(y) \), \(y=\frac{x-\xi }{\lambda }\) in the inner approximation to eliminate the radial part of \(S(u_1({{\textbf {p}}})) \) (defined in (4.2)), which we define as
With this definition
and the terms not in \(S(u_1)\) correspond to \( \frac{\alpha }{\lambda ^3} {\dot{\xi }} \cdot {\nabla } _y U (y) \, \chi + \frac{\alpha }{\lambda ^2 \sqrt{t}} U(y)\) which are in mode 1.
Then we want \(\phi _0^i\) to be an appropriate solution to the equation
where L is the linear operator (5.4), \(t>t_0\) is regarded as a parameter, \(W_2(y)\) is a fixed smooth radial function with compact support, and
By Lemma 5.2 and Proposition 5.1, the choice \({{\textbf {p}}} = {{\textbf {p}}}_0\) is so that (5.22), (5.23) hold. Since the difference between \(S(u_1)\) and \(S_0(u_1)\) contains terms in mode 1 only, we get from Corollary 5.1
In (5.32) we select \(c_0(t)\) such that
and thanks to (5.34) we have
Note that we have
which follows from the constant mass in time of \(u_1({\textbf{p}}_0)\) in (5.22) and the form of the operator \(S_0\) (5.31).
We let \(\phi ^i_0\) be the solution to (5.32) constructed in Lemma 5.1. By (5.15) and (4.5)
and
5.3 Reformulation of the system
In the outer problem (5.9) we would like to separate the effect of the initial condition from the coupling \(G(\phi ^i,\varphi ^o,{\textbf{p}})\).
We take the initial condition in (5.9) to be
and let \(\varphi ^*(x,t)\) denote the solution of
The initial condition \(\varphi ^*_0(x)\) will be later used to prove the stability claimed in Theorem 1.1. The topology for \(\varphi _0^*\) will be specified later on.
Note that \(\nabla _x \Gamma _0 (\frac{x-\xi }{\lambda })= -4 \frac{x-\xi }{|x-\xi |^2 +\lambda ^2}\) so that \(\varphi ^* \) is a function of the parameters \(\lambda ,\xi \). Therefore we will write \( \varphi ^*(x,t;{{\textbf {p}}})\) when convenient.
We decompose
where
with \(\lambda _0\) the function constructed in Proposition 5.1 and \(\alpha _0\) chosen so that (5.22) holds.
We substitute the expressions for \(\phi ^i\), \(\varphi ^o\) and \({{\textbf {p}}}\) in (5.38) into the Eqs. (5.5), (5.9), and are led to the following problem for \(\phi \), \(\varphi \)
where \({\tilde{\chi }}\) is defined in (5.8),
\(\delta >0\) is a small constant to be fixed later on, and \(\chi _0\) is as in (2.5). We recall that F and G are defined in (5.6) and (5.10). The expressions for \(F_2\) and \(G_2\) depend on the initial condition \(\varphi ^*_0\) through \(\varphi ^*\) (5.37) and \(\phi _0\). The role of \(\phi _0\) will be clarified later on.
By the estimate for \(\ddot{\lambda }_0\) in Proposition 5.1 and (5.35) we get
The reason that we introduce the cut-off \({\tilde{\chi }}_2\) is to achieve
if \(\nu < 1 +2\delta - \frac{\sigma }{2}\). We will choose \(\delta \) and \(\sigma \) positive small numbers such that \(2\delta -\frac{\sigma }{2}>0\) so that we can find \(1<\nu < 1 +2\delta - \frac{\sigma }{2}\).
5.4 Splitting the inner solution \(\phi \)
We perform one more change in the formulation (5.39), (5.40), which consists in decomposing
The function \(\phi _1\) will solve an equation with part of the right hand side of (5.39), which will be projected so that it satisfies the zero second moment condition.
For any h(y, t) with sufficient spatial decay we define
and
which denote the mass, second moment and center of mass of h.
Let \( W_0 \in C^\infty ({\mathbb {R}}^2)\) be radial with compact support such that
Let \(W_{1,j}\), \(j=1,2\) be a smooth functions with compact support and with the form \(W_{1,j} (y) = {{\tilde{W}}}(|y|) y_j\) so that
We recall that \(W_2\) defined in (5.33).
Then, \(h - m_0[h]W_0\) has zero mass, \(h-m_2[h]W_2\) has zero second moment, and \(h - m_{1,1}[h] W_{1,1} - m_{1,2}[h]W_{1,2}\) has zero center of mass.
We modify of the operator \(B_0\) appearing in (5.39), and defined in (5.7). The idea is to work with a variant of it, which coincides with it for radial functions, but for functions without radial part it is cutoff outside the region \(|y|\lesssim \frac{\sqrt{t}}{\lambda }\). More precisely, we decompose \(\phi \) in a radial part \([\phi ]_{rad}\) defined by
and a term with no radial mode \(\phi _1 = \phi - [\phi ]_{rad}\). We note that the other linear terms in the equation behave well with this decomposition. Then we define
where \(\chi _0\) is a smooth cut-off in \({\mathbb {R}}\) with \(\chi _0(s) = 1\) for \(s\le 1\) and \( \chi _0(s)=1\) for \(s \ge 2\).
With these definitions we introduce the following system for \(\phi _1\), \(\phi _2\), \(\varphi \), \({{\textbf {p}}}_1\),
where
In (5.48) \(\mu _j(t)\) are functions so that the right hand side has center of mass equal to zero. A solution \(\phi _1\), \( \phi _2\), \(\varphi \) to (5.48), (5.49) and (5.50) gives a solution to the system (5.39), (5.40) provided \({{\textbf {p}}}_1\) is such that the following equations are satisfied
5.5 Mass and second moment
In this section we derive some formulas for the mass and second moment appearing in the right hand side of (5.48).
In the computation of \(m_0[ F_3(\phi ,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) ]\) and \(m_2[ F_3(\phi ,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) ]\), the following formulas will be useful.
Lemma 5.3
We have
and
where \(e_1(s)\) is defined by
Recall that \(\Upsilon \) is given in (2.7) and note that
Proof
For this we recall that (c.f. (2.8))
so
Therefore
\(\square \)
Lemma 5.4
We have
Proof
We have defined the second moment \(m_2\) (5.44) integrating with respect to y. Note that
and therefore
We have by Lemma 5.2,
where E, \({{\tilde{E}}}\) are defined in (3.10), (3.11). Let
Since
by (5.22), we have
Replacing m in (5.54) we get
Also under (4.1) we have by (5.24):
where
Combining (5.55), (5.53) and (5.56) we get
We can apply this formula to \({\textbf{p}} = {\textbf{p}}_0\) and get
Note that
because
Therefore,
\(\square \)
6 Proof of Theorem 1.1
Next we define norms, which are suitably adapted to the terms in the inner linear problems (5.48), (5.49). Let us write the linearized versions of these problems as
Given positive numbers \(\nu \), p, \(\varepsilon \) and \(m\in {\mathbb {R}}\), we let
We also defie
We develop a solvability theory of problem (6.1) that involves uniform space-time bounds in terms of the above norms. We will establish two results: one in which the solution “loses” one power of t on bounded sets with respect to the time-decay of h, under radial symmetry and the condition of spatial average 0 at all times. Our second result states that for a general h this loss is only \(t^{\frac{1}{2}} \) if in addition the center of mass and second-moment of h are zero at all times.
For the first result we introduce a parameter in the problem in order to get a fast decay of the solution:
where \({{\tilde{Z}}}_0\) is defined as
where \(m_{Z_0}\) is such that
Proposition 6.1
Assume (4.1). Let \(\sigma >0\), \(\varepsilon >0\) with \(\sigma +\varepsilon <2\) and \(1<\nu <\frac{7}{4}\). Let \(0<q<1\). Then there exists a number \(C>0\) such that for \(t_0\) sufficiently large and all radially symmetric \(h=h(|y|,t)\) with \(\Vert h\Vert _{0,\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
there exists \(c_1 \in {\mathbb {R}}\) and solution \(\phi (y,t) = {\mathcal {T}}^{i,2}_{{{\textbf {p}}}} [h]\) of problem (6.3) that defines a linear operator of h and satisfies the estimate
Moreover \(c_1\) is a linear operator of h and
We also consider the problem
where the function \(W_{1,j}\) have been defined in (5.45).
Proposition 6.2
Assume (4.1). Let \(0<\sigma <1\), \(\varepsilon >0\) with \(\sigma +\varepsilon <\frac{3}{2}\) and \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{5}{4})\). Let \(0<q<1\). Then there is C such that for \(t_0\) large the following holds. Suppose that h satisfies \(\Vert h\Vert _{0,\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
Then there exists a solution \(\phi (y,t)\), \(\mu _j(t)\) of problem (6.5) that defines a linear operator of h and satisfies
The parameters \(\mu _j\) satisfy
where \({\tilde{\mu }}_j\) are linear functions of h with
We denote this solution by \(\phi = {\mathcal {T}}^{i,1}_{{{\textbf {p}}}} [h]\).
The proof of the Propositions 6.1 and 6.2 is divided into different steps and presented in Sects. 8–12.
Next we consider the linear outer problem:
where
For a given function g(x, t) we consider the norm \(\Vert g \Vert _{**,o}\) defined as the least \(K\geqq 0\) such that for all \((x,t)\in {\mathbb {R}}^2\times (t_0,\infty )\)
Accordingly, we consider for a function \(\phi ^o(x,t)\) the norm \(\Vert \phi \Vert _{*,o}\) defined as the least \(K\geqq 0\) such that
for all \((x,t)\in {\mathbb {R}}^2\times (t_0,\infty )\).
We assume that the parameters \(a,b,\beta \) satisfy the constraints
Proposition 6.3
Assume that the parameter functions \({{\textbf {p}}} =(\lambda , \alpha ,\xi )\) satisfy conditions (4.1) and the numbers \(a,b,\beta \) satisfy (6.9). Then there is a constant C so that for \(t_0\) sufficiently large and for \(\Vert g\Vert _{**,o}<\infty \), there exists a solution \(\phi ^o= {\mathcal {T}} ^o_{{{\textbf {p}}}}[g ]\) of (6.6) with \(\phi ^o_0=0\), which defines a linear operator of g and satisfies
For the initial condition \(\phi _0^o\) in (6.6) we consider the norm \(\Vert \varphi _0^o \Vert _{*,b}\) defined as
We have an estimate for the solution of (6.6) with \(g=0\) and \(\Vert \phi ^o_0 \Vert _{*,b}<\infty \).
Proposition 6.4
Assume that the parameter functions \({{\textbf {p}}} =(\lambda , \alpha ,\xi )\) satisfy conditions (4.1) and the numbers \(a,b,\beta \) satisfy (6.9). Then there is a constant C so that for \(t_0\) sufficiently large and for \(\Vert \phi ^o_0 \Vert _{*,b}<\infty \) there exists a solution \(\phi ^o\) of (6.6), which defines a linear operator of \( \phi ^o_0\) and satisfies
The proofs of Propositions 6.3 and 6.4 are contained in Sect. 13.
In what follows we work with \({{\textbf {p}}}_1 \) of the form
that is, we take \(\lambda = \lambda _0\), \(\alpha = \alpha _0+\alpha _1\), \(\xi = \xi _1\), where \(\lambda _0\) and \(\alpha _0\) have been fixed in Sect. 5.1, and we write
Next we define suitable operators that allow us to formulate the system of equations (5.48), (5.49), (5.50), and (5.52) as a fixed point problem. We let
Then the equations (5.48), (5.49),(5.50) can be written as
Next we consider the equations (5.52), that is, \(m_0[F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*)] (t)\equiv 0\) and \(\mu _j(t) \equiv 0\). By (5.51) and (5.41)
and using Lemma 5.3,
This motivates the definition
Similarly, by (5.51) and (5.41), asking that \(\mu _j\equiv 0\) in (5.48) is equivalent to
This motivates the definition
Then we define \({\mathcal {A}}_p\) by
Then
is equivalent to the equations (5.52).
We write
and
and the objective is to find \(\vec {\phi }\) such that
The operator \({\mathcal {A}}\) depends on the initial condition \(\varphi _0^*\) appearing in the parabolic problem (5.37), and we will stress its dependence later on when proving the stability assertion in Theorem 1.1.
We define the spaces on which we will consider the operator \({\mathcal {A}}\) to set up the fixed point problem. For certain choices of constants \(\nu \), q, \(\sigma \), \(\varepsilon \), a, b, \(\beta \), \(\gamma \), \(\Theta \) that we will make precise later, we let
where the norms \(\Vert \phi \Vert _{1,\nu -\frac{1}{2},\frac{q-1}{2},4,2+\sigma +\varepsilon }\) and \(\Vert \varphi \Vert _{*,o}\) are defined in (6.2), (6.7) and \(\Vert \xi _1\Vert _{C^1,\mu ,m}\) is defined by
for a function \(g \in C^1([t_0, \infty ))\).
We choose in the definition of the outer norm (6.8)
With these choices we see that (6.9) are satisfied. Also \(\nu \) will be in the range \(1<\nu <\frac{3}{2}\) so the interval for b is not empty in (6.14).
We use the following notation: for \({{\textbf {p}}}_1= (0,\alpha _1,\xi _1)\),
and for \( \vec {\phi } = ( \phi _1,\phi _2, \varphi , {{\textbf {p}}}_1) \),
With the above notation, given \(\varphi _0^*\) with \(\Vert \varphi _0^*\Vert _{*,b} \) sufficiently small, we consider the fixed point problem
with \(\vec {\phi } \) in a suitable close ball of X. A solution of this fixed point problem yields a solution of the system of Eqs. (5.48), (5.49), (5.50), (5.52), which in turn gives a solution to (3.1).
We claim that for some constant C independent of \(t_0\gg 1\), if \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \), and \(\Vert \vec {\phi }\Vert _X\le 1\), then
for some \(\vartheta >0\) small, a constant C independent of \(t_0\), and \(t_0\) sufficiently large.
Indeed, by Proposition 6.2 we have
We recall the expansion of \(F_3\) in (5.51). To estimate \( E_2 {\tilde{\chi }}_2\) we use (5.43) to get
where \(\delta \), \(\sigma \) are positive small constants and are assumed to satisfy \(2\delta -\frac{\sigma }{2}>0\). Then we take \(\nu \) in the range
with \(\nu \) close to 1.
Let us consider the term \( \lambda ^4 [ S_0({{\textbf {p}}}_0 + {{\textbf {p}}}_1) - S_0({{\textbf {p}}}_0)]\) in \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \) (c.f. (5.51)). The formula \( \lambda ^4 [ S_0({{\textbf {p}}}_0 + {{\textbf {p}}}_1) - S_0({{\textbf {p}}}_0)]\) (c.f. (5.31)) contains for example the term, evaluated at \(y = \frac{x-\xi _1}{\lambda _0}\),
But
so
for some \(\vartheta >0\).
Similarly,
so
for some \(\vartheta >0\). The last term in the expression (6.19) is similar.
The terms in \( \lambda ^4 [ S_0({{\textbf {p}}}_0 + {{\textbf {p}}}_1) - S_0({{\textbf {p}}}_0)]\) that contain the function \(\varphi _{\lambda _0}\) are
In \( {\lambda _0}^4 [ S_0({{\textbf {p}}}_0 + {{\textbf {p}}}_1) - S_0({{\textbf {p}}}_0)]\) these terms appear evaluated at y and then at \(\frac{\xi _1}{\lambda _0}+y\). Using estimates for the the second derivative of \(\varphi _{\lambda _0}\) similar to Lemma 4.1 and assuming
we get
The main term in \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \) that depends on the outer solution is \( \lambda ^2 U \varphi ^o \) with \(\varphi ^o = \varphi ^* + \varphi \) defined in (5.38). Then we have
Therefore
Regarding the function \(\varphi ^*\) (c.f. (5.37)) we note that it has the estimate
by Proposition 6.4, provided (6.9) holds, and therefore
Let us analyze some of the terms in \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \) that depend on the inner solutions \(\phi _1\) and \(\phi _2\). For instance
We have the estimate
and we get
for some \(\vartheta >0\).
We also have, writing \(\phi = \phi _1+\phi _2\),
for some \(\vartheta >0\), if
Let us estimate the term \( \nabla _y \cdot ( U \nabla _y( {\hat{\psi }} - \psi )) {\tilde{\chi }} \) appearing in (5.6), where \( {\hat{\psi }} = (-\Delta )^{-1} (\lambda ^{-2} \phi ^i \chi )\), \( \psi = (-\Delta )^{-1} (\lambda ^{-2} \phi ^i )\). We recall that \(\phi ^i = \phi ^i_0 + \phi \), c.f. (5.38), and therefore we can decompose \({\hat{\psi }} = {\hat{\psi }}^i_0 + {\hat{\psi }}_1\) where \({\hat{\psi }}^i_0 = ( -\Delta )^{-1}( \lambda ^{-2} \phi ^i_0 \chi )\) and \({\hat{\psi }}_1 = (-\Delta )^{-1} ( \lambda ^{-2} \phi \chi )\). Similarly, we can decompose \(\psi = \psi ^i_0 + \psi _1\) where \(\psi ^i_0 = ( -\Delta )^{-1}( \lambda ^{-2} \phi ^i_0 )\) and \(\psi _1 = (-\Delta )^{-2} ( \lambda ^{-1} \phi )\). By linearity we need to estimate separately \( \nabla _y \cdot ( U \nabla _y( {\hat{\psi }}^i_0 - \psi ^i_0)) \) and \(\nabla _y \cdot ( U \nabla _y( {\hat{\psi }}_1 - \psi _1)) \). Let us consider the latter one. Note that
From the definition of the norm \(\Vert \phi \Vert _{1,\nu -\frac{1}{2},\frac{q-1}{2},4,2+\sigma +\varepsilon }\)
and so
Then
This and a similar estimate for \(U \phi ( 1-\chi ) \) give
for some \(\vartheta >0\). A similar estimate is obtained for \(\Vert \nabla _y \cdot ( U \nabla _y({\hat{\psi }}^i_0 - \psi ^i_0)) {\tilde{\chi }} \Vert _{0,\nu ,6+\sigma ,\varepsilon }\) using (5.36).
Let us estimate next the term \(\lambda ^2 \nabla _y \cdot (\varphi _\lambda \nabla _y \psi ) {\tilde{\chi }}\), where we recall, \(\psi = (-\Delta )^{-1} (\lambda ^{-2} \phi )\). To do this we use that \(\phi = \phi _1+\phi _2\) has zero mass and center of mass, that is,
This and the estimate (6.23) imply
by an argument similar to Remark 9.1. On the other hand, from (4.3)
Therefore
From this coupled with a similar estimate for \(\lambda ^2 \varphi _\lambda \phi \) we get
for some \(\vartheta >0\).
The remaining terms in \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \) are estimated in a similar way and we get the validity (6.17).
Proceeding in the same way we get a Lipschitz bound. Assuming \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \), for \(\Vert \vec {\phi }_1\Vert _X \le 1\) and \(\Vert \vec {\phi }_2\Vert _X\le 1\) we have
for some \(\vartheta >0\) small, a constant C independent of \(t_0\), and \(t_0\) sufficiently large. Indeed, the Lipschitz estimate with respect to \(\phi _1\), \(\phi _2\), and \(\varphi \) is direct from the explicit dependence of \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \) on these variables, which is either linear or quadratic. The Lipschitz dependence on \(\xi _1\) (where \({{\textbf {p}}}_1 = ( \alpha _1,\xi _1))\) is also direct from the explicit form of \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \). The Lipschitz condition with respect to \(\alpha _1\) appears as an explicit dependence on this variable in \(F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) \).
Let us estimate the operator \({\mathcal {A}}_{i2}\). We claim that if \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \) and \(\Vert \vec {\phi }\Vert _X\le 1\), then
Indeed, we apply Proposition 6.1 to get
and since \(W_2\) has compact support,
Using the definition of \(F_3\) (5.51)
We have by (5.43) (assuming \(\sigma <\frac{1}{2}\)),
Therefore, asking that
we get
for some \(\vartheta >0\).
By (5.41)
Of these terms, the largest is the first one. By Lemma 5.4, and since \(\lambda = \lambda _0\), we get
But
under the assumption
The second term in (6.26) is much smaller. For the last term in (6.26) we use Lemma 5.3 and (5.29), (5.30) to get
and therefore
Combining (6.26), (6.27) and (6.29) we get
Let’s estimate the remaining terms in \(m_2[ F_3(\phi ,\varphi ,{{\textbf {p}}}_1,\varphi _0^*) ]\). Consider
which appears in the definition of F, where \(\phi = \phi _1 + \phi _2\). Let us recall that \(\psi _\lambda =(-\Delta _x)^{-1}\varphi _\lambda \) and let’s write
Integrating by parts,
Using the Pohozaev type identity
and integrating by parts, we get
Therefore
Using that \(\psi = (-\Delta )^{-1} \phi \), and
we have (see Remark 9.1) for any \(\varrho >0\) small,
Using that \(\varphi _\lambda \), and \(\psi _\lambda \) are radial and
by Lemma 4.1 we have
Then
Let us consider the contribution of the term \(\lambda ^2 U \varphi ^*\). Thanks to (6.21)
under the condition
The other terms in \(m_2\) are estimated in a similar way and we get (6.24).
Similarly we get that if \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \), then for \(\Vert \vec {\phi }_1\Vert _X \le 1\) and \(\Vert \vec {\phi }_2\Vert _X\le 1\) we have
for a constant C independent of \(t_0\), where \(t_0\) sufficiently large.
Let us estimate the operator \( {\mathcal {A}}_o[\phi _1,\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*]\). We claim that if \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \), then for \(\Vert \vec {\phi }\Vert _X\le 1\),
and for \(\Vert \vec {\phi }_1\Vert _X\le 1\), \(\Vert \vec {\phi }_2\Vert _X\le 1\) and \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1 \),
Note that \(\frac{q+1}{2}-\beta >0\) by (6.14).
Indeed, by Proposition 6.3
where we recall \(G_2\) defined in (5.42).
We start with the term \( \lambda ^{-4} E_2 (1-{\tilde{\chi }}_2)\chi \). Using the estimate (5.43) we get
for some \(\vartheta >0\) provided
We also directly get from (4.6)
for some \(\vartheta >0\) if \(a<4\).
Regarding the terms in G (c.f. (5.10)) that the depend linearly on \(\phi ^i = \phi ^i_0 + \phi \), we have for \(\Vert \phi \Vert _{1,\nu -\frac{1}{2},\frac{q-1}{2},4,2+\sigma +\varepsilon } < \infty \)
which implies
since \(\beta <\frac{q+1}{2}\), which is one of the conditions in (6.14).
We also have, using (5.36),
for some \(\vartheta >0\) if
A similar estimate holds for the other terms depending on \(\phi ^i\).
Some of the terms in G that depend on \(\varphi ^o = \varphi ^* + \varphi \) are
which implies that
by Proposition 6.4. Other terms are estimated in a similar way.
Let us estimate the operator \({\mathcal {A}}_p\), which is defined by the equations (6.13). We claim that if
and \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1\), \(\Vert \vec {\phi }\Vert _X \le 1\), \(\vec {\phi } = (\phi _1,\phi _2,\varphi ,{\textbf{p}}_1)\), then
for some \(\vartheta >0\). Similarly, we have the following Lipschitz estimate. If \( t_0^{a-1} (\log t_0)^\beta \Vert \varphi _0^* \Vert _{*,b}\le 1\), then for some \(\vartheta >0\), and for \(\Vert \vec {\phi }_1\Vert _X\le 1\), \(\Vert \vec {\phi }_2\Vert _X\le 1\),
for some \(\vartheta >0\).
Indeed, by (6.11)
where
Using (5.43) and \(\int _{{\mathbb {R}}^2} E_2 dy=0\) we get
This gives
under the assumption
The largest contribution in \(I_2\) comes from the term \(\lambda ^2 U \varphi ^o\) in \(F(\phi ^i_0 + \phi , \varphi ^* + \varphi ,{{\textbf {p}}}_0+{{\textbf {p}}}_1) \) (c.f. (5.6)). The estimate of this term is
and so
under the assumption
Similar estimates for the remaining terms give
Regarding \(I_3\), using (4.5) we have
Putting together (6.35), (6.38), and (6.39) we get
assuming also that
The computations leading to (6.33) are very similar, under the assumption
This restriction arises when considering the largest term in the expression (6.12), namely comes from estimating the term \(\lambda _0^2 m_{1,j}[\varphi _{\lambda _0} \phi {\tilde{\chi }}] \) (\(\lambda _0^2 \varphi _{\lambda _0} \phi \) is one of the terms in (5.6))
Let us summarize the restrictions on the parameters. We let \(0<q<1\) be fixed. We take
and
because of (6.18), (6.20), (6.25). We also need
by (6.28), (6.37) and by (6.30) and (6.14). We take
Together with the above inequalities we want also the relations \( \sigma +\varepsilon <2\), \(\nu +\frac{1}{2}<\frac{7}{4}\) for Proposition 6.1 and \(\sigma +\varepsilon <\frac{3}{2}\), \(\nu <\min ( 1 + \frac{\varepsilon }{2}, 3-\frac{\sigma }{2},\frac{5}{4}) \) for Proposition 6.2. The condition (6.9) for Propositions 6.3 and 6.4 hold by (6.14). We see that all these restrictions are satisfied by choosing first \(\delta \), \(\sigma >0\) small so that \(2\delta - \frac{\sigma }{2}>0\). Then we take \(\nu >1\) close to 1, then let \(a=\nu +\frac{5}{2}\) and b satisfying (6.14). Then \(\Theta \), \(\beta \) and \(\gamma \) can be selected. Note that with the above procedure we are getting the restriction \(b>5\).
We already have all elements to solve the fixed point problem (6.16), which we recall
where \({\mathcal {B}}\) is the closed unit ball in the Banach space of functions \(\vec {\phi } \) with \(\Vert \vec {\phi }\Vert _X < +\infty \) and the norm defined in (6.15). Thus
Let \(\varphi _0^*\) be such that \(t_0^{a-1} (\log t_0)^{\beta } \Vert \varphi _0^*\Vert _{*,b}\le 1\). Estimates (6.17), (6.24), (6.31) and (6.34), imply that, enlarging the parameter \(t_0\) if necessary, \({\mathcal {A}}\) maps \({\mathcal {B}}\) into itself. We also get that \({\mathcal {A}}\) is a contraction mapping on \({\mathcal {B}}\). The contraction mapping principle yields the existence of a unique fixed point in \({\mathcal {B}}\), which then yields the required existence result.
6.1 Stability
Theorem 1.1 gives that if \(\varphi _0^*\) has mass zero and is small so that \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert \varphi _0^*\Vert _{*,b}\le 1\), then the function
solves (3.1) and blows-up in the way described in Theorem 1.1. This follows from the form of the ansatz (3.2), (3.13), (5.1), (5.38), where \(\phi =\phi _1+\phi _2\), and \(\phi _1\), \(\phi _2\), \(\varphi \), \(\varphi ^*\) satisfy respectively the equations (5.48), (5.49), (5.50) and (5.37). The initial value of u is
We recall that \({\tilde{\varphi }}_\lambda \) is defined in (3.9). The function \({\tilde{\varphi }}\) doesn’t depend on \(\xi \) and is radial about the origin.
We let \(u_0^*(x) = u^*(x;0)\). Note that \(u_0^*\) is radial and so its center of mass is zero.
To prove stability, we would like to prove the following intermediate step: given v defined on \({\mathbb {R}}^2\) small, with mass zero and under some additional assumptions to be defined later on, we would like to find \(\varphi _0^*\) with mass zero such that
The equation (6.42) for \(\varphi _0^*\) has the form
Computing the mass we find that \(\alpha (t_0;\varphi _0^*)=\alpha (t_0;0)\). Note that \(\lim _{t\rightarrow \infty }\xi (t)=0\) by (6.12). Then the center of mass of \(u(\cdot ,t)\) satisfies
Since the center of mass is preserved
Let’s assume that the center of mass of v and \(\varphi _0^*\) are both zero. Then, computing the center of mass we find that
Then the Eq. (6.43) reduces to
We will prove at the end of this section the following.
Proposition 6.5
There is \(\delta >0\) so that if \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert v\Vert _{*,b}\le \delta \), v has mass and center of mass equal to zero, then
is equivalent to
To prove stability we first observe that if \(v:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) satisfies \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert v\Vert _{*,b}\le \delta \), has mass zero, and
then \(u_0^*+v = u^*(\varphi _0^*) \) for \(\varphi _0^*=v\), by Proposition 6.5.
Now consider a general v with \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert v\Vert _{*,b}\le \delta \) (for a possibly smaller \(\delta >0\)), and mass zero. We want to show that the initial condition \(u_0^*+v\) produces a solution to (3.1) with infinite time blow as described in Theorem 1.1. Consider
where \(p\in {\mathbb {R}}^2\) and \(\Lambda >0\). Note that \(u_{\Lambda ,p}\) has mass \(8\pi \). Then we select \(\Lambda \) and p such that
Note that \(|\Lambda ^2-1| \le Ct_0^2 \Vert v\Vert _{*,b}\ll 1\) and \(|p|\le C t_0 \Vert v\Vert _{*,b}\ll 1\). Then we expand
and w satisfies \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert w\Vert _{*,b}\le C \delta \), has mass zero, center of mass zero and second moment equal to 0. By the previous claim, the initial condition \(u_{\Lambda ,p} (x) = u_0^* + w\) is such that the solution to (3.1) blows up as in Theorem 1.1. Then the same is true for the initial condition \(u_0^*+v\) after a scaling and translation in space.
6.2 Proof of Proposition 6.5
Lemma 6.1
Assume that \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert v\Vert _{*,b}\le 1\), that v has mass and center of mass equal to zero, and that
Then
Proof
From (6.46), \(\varphi _0^*=v\) solves (6.45), and therefore \(u_0^*+v\) is an initial condition for (3.1) for which the solution blows up in infinite time. The solution u to (3.1) preserves the second moment:
We compute the expansion of \(\int _{{\mathbb {R}}^2} u(x,t)|x|^2dx\) as \(t\rightarrow \infty \), based on the expression (6.41).
Note that \(\lim _{t\rightarrow \infty }\xi (t)=0\) by (6.12). Then
as \(t\rightarrow \infty \). By explicit computation
as \(t\rightarrow \infty \).
Using Lemma 4.1
Using also Lemma 4.1 to estimate the mass and first moment of \(\varphi _{\lambda _0}\) we get
Using (6.47), (6.48) and the estimates for \(\phi _0^i\) (5.36), \(\phi =\phi _1+\phi _2\) that arise from \( \Vert \phi \Vert _{1,\nu -\frac{1}{2},\frac{q-1}{2},4,2+\sigma +\varepsilon }<\infty \), and \(\varphi \), \(\varphi ^*\) which arise from \(\Vert \varphi \Vert _{*,o}<\infty \), \(\Vert \varphi ^*\Vert _{*,o}<\infty \), we get that
as \(t\rightarrow \infty \). But \(\lambda _0\) was constructed in Proposition 5.1 with the expansion
as \(t\rightarrow \infty \), where \(c_0>0\) is a constant. Therefore
and evaluating at \(t=t_0\) we obtain
We can apply the previous calculation to \(v=0\) and arrive at
This shows that
\(\square \)
We need an expansion for \(c_1(\varphi _0^*)-c_1(0) \).
Lemma 6.2
Assume that \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert \varphi _0^*\Vert _{*,b}\le 1\) and that \(\varphi _0^*\) has mass and center of mass equal to zero. Then
where \(a_0\not =0\) and \(R_0\) satisfies
Proof
In the following calculations \(\lambda =\lambda _0\).
First we need to estimate the Lipschitz constant of the solutions \(\phi _1\), \(\phi _2\), and \(\varphi \) with respect to \(\varphi _0^*\). We claim that
We discuss briefly the proof of these estimates. One of the main terms in the right hand side of (5.50), written for the difference \(\varphi (\varphi _0^*)-\varphi (0)\) is
which implies
since \(\beta <\frac{q+1}{2}\), which is one of the conditions in (6.14). (Here \(\chi \) depends on \(\varphi _0^*\). There is another ther in the difference that depends on \(\chi (\varphi _0^*)-\chi (0)\) and is estimated similarly.) Then
Considering \(\varphi \) as an operator of \(\phi \) we examine the effect of the therm \(\lambda ^2 U \varphi \). This term appears in the right hand side of (5.48), where the effect is less important, and in the computation of \(\alpha _1\). Estimating the right hand side of (5.50) as in (6.32), using Proposition 6.3 gives that
We consider now the effect of \(|\alpha _1[\phi ](t)|\) in the right hand side of (5.49), where thanks to Lemma 5.4 appears mainly as \( \alpha _1(t) W_2(y)\), where \(W_2\) is radial with compact support. Then Proposition 6.1 gives
Then
The estimate for \(\phi _1\) is actually better, and therefore
This implies (6.50). Replacing this in (6.53) we obtain (6.51), and similarly we get (6.52).
The parameter \(c_1\) appears in the second inner equation in (5.49), which we write as
where
Note that \(\phi _2\) in (6.54) is radial, so the operator B defined (5.47) reduces to \(B[\phi ]=\lambda {\dot{\lambda }} ( 2 \phi + y \cdot \nabla \phi ) = \lambda {\dot{\lambda }} \nabla \cdot (y\phi )\). Multiplying by \(|y|^2\) and integrating on \({\mathbb {R}}^2\) gives
Then
But \(\phi _2(y,t_0)=c_1 {{\tilde{Z}}}_0(y)\) so
In particular
The function \(h(t,\varphi _0^*) =m_2[F_3(\phi _1+\phi _2,\varphi ,{{\textbf {p}}}_1,\varphi _0^*)] (t)\) is analyzed near (6.25). We follow the same steps. Using the definition of \(F_3\) (5.51)
We note that \({\tilde{\chi }}\), \({\tilde{\chi }}_2\) also depend on \(\varphi _0^*\) because \(\xi \) depends on \(\varphi _0^*\). By (5.41)
where
The main term is \(I(\varphi _0^*)\) and the others are treated as perturbations.
By Lemma 5.4, since \(\lambda = \lambda _0\), we get
where
By (6.11)
where
Let
so that
By (5.38), \(\varphi ^o=\varphi ^*+\varphi \), where \(\varphi =\varphi (\varphi _0^*)\) solves (5.50) and \(\varphi ^*\) solves (5.37). Therefore
where
Integrating (5.37) on \({\mathbb {R}}^2\) we find that
and therefore
Then from (6.57)
Using this and (6.56) we get
Hence
where
From (6.55) it follows that
where
We can relate the integral \(\int _{t_0}^\infty \int _{{\mathbb {R}}^2}\varphi ^*(x,s,\varphi _0^*) dxds\) with the second moment of \(\varphi _0^*\) as follows. We multiply the equation of \(\varphi ^*\) (5.37) by \(|x-\xi (t)|^2\) and integrate on \({\mathbb {R}}^2\) to get
But
and
Using the explicit expressions for U and \(\Gamma _0\) and writing \(y=\frac{x}{\lambda }\), \(\rho =|y|\), we get
So
and we find that
Integrating and using (6.58) we find that
by (6.44), where
We claim that \(R_0(\varphi _0^*)\) satisfies (6.49). Indeed, let us look at
Similarly (6.36), we have
Similar computations for the other terms of \(I_0\) give
It follows that
The other terms in \(R_0\) are estimated similarly. \(\square \)
Proof of Proposition 6.5
If \(t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \Vert v\Vert _{*,b}\le 1\) and \(c_1(v)-c_1(0)=0\), then Lemma 6.1 implies that \(\int _{{\mathbb {R}}^2} v(x)|x|^2dx=0\).
To prove the converse, let
so that \(\Vert v_1\Vert _{*,b}=1\) (norm defined in (6.10)). Assuming \(\mu t_0^{\nu +\frac{3}{2}} (\log t_0)^{\beta } \le \delta \) and \(\delta >0\) small, we have by Lemma 6.2
for some constant \(c\not =0\). Note that is \(c_1(\varphi _0^*)\) continuous function of \(\varphi _0^*\), and so is \(R_0(\varphi _0^*)\). By the intermediate value theorem, there is \(\mu = O(t_0 \Vert v\Vert _{*,b})\) such that \(c_1(v+\mu v_1)-c_1(0)=0\). By Lemma 6.1\(\int _{{\mathbb {R}}^2}(v(x) + \mu v_1(x))|x|^2dx=0\), which implies that \(\mu =0\). But then \(c_1(v)-c_1(0)=0\). \(\square \)
7 The mass of \(\varphi _\lambda \)
We devote this section to prove Proposition 5.1. To that purpose, a basic step is to derive a formula for the mass of \(\varphi _\lambda \) defined in (3.12).
Let us write
where \(\varphi _\lambda ^{(1)}\) and \(\varphi _\lambda ^{(2)}\) are the solutions, given by Duhamel’s formula, of the following problems
where the operator \(\Delta _6\) is defined in (3.8) and \({{\tilde{E}}}\) in (3.11). We let \(\varphi [p,\lambda ](r,t)\) be the solution of the problem
given by Duhamel’s formula. By definition, we have
In definitions (7.2), (7.3), (7.4), the parameter function \(\lambda (t)\) is assumed to be defined for \(t>\frac{t_0}{2}\). In the rest of this section we also assume the validity of the condition stated for \(\lambda \) in (4.1), namely
for some fixed constant C. Let us define
In what follows we shall only deal with radial functions on \({\mathbb {R}}^2\) and sometimes we will consider them as radial functions on \({\mathbb {R}}^6\). For a fixed constant \(c_0>0\) we let
The following expansion holds.
Lemma 7.1
Assume that \(\lambda \) satisfies (7.5). Let \(0<\gamma <2\), \(m\in {\mathbb {R}}\) and suppose that \(\Vert p\Vert _{\gamma ,m} <\infty \). Then
where \(R[p,\lambda ]\) satisfies
If \(\lambda _1,\lambda _2\) satisfy
then we also have
For the proof of the above result we will need the following calculation.
Lemma 7.2
Let
Then
Proof
Let \(\varphi _0\) be given by
which solves
Then
Write
Then
and we want q(s) bounded for \(s\rightarrow 0\), \(q(s) = s^{-4}(1+o(1))\) as \(s\rightarrow \infty \). A calculation using the explicit element in the kernel of the linear operator, \(s^{-4}\), gives
and then (7.9) follows. \(\square \)
Proof of Lemma 7.1
The solution \(\varphi [p,\lambda ]\) of (7.4) has the formula
Writing
we have
Using (7.9) we have
Let us notice that
We decompose
where
and separately estimate each term. To estimate \(I_1\) we note that for \(s \le t/2\) we have \(\frac{s}{t-s}\le 1\). Assuming that \(\chi (x) = 0 \) for \(x\ge 2\) we obtain
We estimate for \(s \le t/2\),
where we have used that \(Z_0(\rho ) \le C / \rho ^4\) and \(1-e^{-\frac{z^2}{4}} ( 1+\frac{z^2}{4}) \le C z^4\). Therefore
Let us analyze \(I_2\). We write
where
and
A calculation gives that
Next we find a bound for \(I_{2,a}\). Using that \(Z_0\) is a bounded function and \(| 1-e^{-\frac{z^2}{4}} ( 1+\frac{z^2}{4}) |\le C z^4\), we get
It follows that
Using that \(| 1-e^{-\frac{z^2}{4}} ( 1+\frac{z^2}{4}) |\le C z^4\), we get
and similarly as before,
Using that
we get
But \(\frac{\lambda (s)}{\sqrt{t-s}}\le 2\) in the considered range of s, and then
Finally, for \(I_{2,d}\),
Then
Finally we estimate
In summary, by (7.10) we have written
and each of the expressions \(I_1\), \( I_{2,a}\), \( I_{2,b} \), \( I_{2,c} \), \( I_{2,d} \), \( I_3\) are linear operators of p with the estimate
The proof of (7.8) follows from the explicit expressions for the terms \(I_j\) in R, and similar estimates as before. \(\square \)
Lemma 7.3
Suppose that \(\lambda \) satisfies (7.5) and \(\varphi _\lambda ^{(2)}\) be given by (7.3). Then
as \(t\rightarrow \infty \), where \( O( \frac{1}{t^2 (\log t)^2} ) \) is uniform in \(t_0\). With \(\lambda ^*\) given by (7.7), if \(\lambda _1,\lambda _2\) satisfy
then we also have
Proof
For simplicity of notation let us write \(\varphi (x,t;\lambda ) = \varphi _\lambda ^{(2)}(x,t)\). Let us write the right hand side of equation (7.3) in the following form
To compute \(\varphi (0,t;\lambda )\) let us define the following approximation of it
where \({\tilde{\varphi }}(r,t)\) solves the radial heat equation in dimension 6:
and
The solution \({\tilde{\varphi }}(r,t)\) to problem (7.13) can be expressed in self-similar form as
We find for g the equation
Using that the function \(\frac{1}{\zeta ^4}\) is in the kernel of the homogeneous equation, we find the explicit solution of (7.14),
To find the solution \({\tilde{\varphi }}\) with suitable decay at infinity we let
where
is a second solution of the homogeneous equation, linearly independent of \(\frac{1}{\zeta ^4}\) and
We observe that
which makes the solution (7.15) the only one with decay faster than \(O(\zeta ^{-4})\) as \(\zeta \rightarrow +\infty \). An explicit calculation gives that \( I = -8, \) and therefore
Then, using a barrier for the equation satisfied by \(\varphi (x,t;\lambda ) - {\hat{\varphi }}(x,t)\) we get
for \(t\ge 2\), where \(0<c<\frac{1}{4}\). From (7.16) and (7.17) we obtain (7.11).
The proof of (7.12) is similar. \(\square \)
Lemma 7.4
Suppose that \(\lambda \) satisfies (7.5) and \(\varphi _\lambda ^{(2)}\) be given by (7.3). Then
where \(\Upsilon \) is defined in (2.7), that, is, \(\Upsilon = \int _0^\infty (\chi _0(s)-1) s^{-3}ds\).
Proof
Integrating (7.3)
From (7.11)
and we compute
where \(s = \frac{r}{\sqrt{t}}\). Then
Therefore
and integrating we get
This is the desired expansion (7.18). \(\square \)
As a corollary from Lemma 7.1 and Lemma 7.4 we get:
Corollary 7.1
Assume \(\lambda \) satisfies (7.5). Then
where R is as in Lemma 7.1.
Lemma 7.5
Let \({{\tilde{E}}}\) be defined by (3.11). Assume that \(\lambda \) satisfies (7.5). Then
Proof
Similarly to the proof of Lemma 7.4 we have
where \( r = |x|\), \(s = \frac{r}{\sqrt{t}}\), and so
This is (7.19). \(\square \)
Lemma 7.6
Let E be defined by (3.10). Assume that \(\lambda \) satisfies (7.5). Then
Proof
We have from (3.10)
and we have already computed \(\int _{{\mathbb {R}}^2} {{\tilde{E}}} |x|^2 dx\) in (7.19). We have
and so
\(\square \)
7.1 Proof of Proposition 5.1
Let
For the proof we proceed by linearization, that is we look for a function \(\lambda _0\) satisfying
with the expansion
where \(\lambda ^*\) was defined in (7.7), that is, \(\lambda ^*(t) = \frac{c_0 }{\sqrt{\log t}}\) and \({\tilde{\lambda }}_0(t)\), \(t>\frac{t_0}{2}\), is a correction. Here \(c_0>0\) is a fixed constant.
We claim that
with C independent of \(t_0\). In the rest of the proof C will be a constant independent of \(t_0\) (for \(t_0\) large).
Indeed, using the decomposition (7.1) and the notation (7.4) we have
and
By Lemma 7.1 we have
Therefore
On the other hand, by Lemma 7.4 we have
and by Lemma 7.5
Using the explicit form of \(\lambda ^* \) and the previous formulas we deduce (7.20).
Next let us rewrite slightly the operator \(I [\lambda ]\) as follows. We have
Let us define
This is similar to the decomposition given in Lemma 7.1, but we have changed the interval of integration to \([\frac{t}{2},t-\lambda ^*(t)^2]\). We decompose the integral
where \(0<\vartheta <\frac{1}{2}\) is a fixed constant.
We change variables \(\mu = \lambda ^2\), so that
Let \(\eta \) be a smooth cut-off such that \(\eta (t) = 0 \) for \(t<\frac{3}{4}t_0\), \(\eta (t) =1 \) for \(t>t_0\). We define
which we write
where
Note that \(I[\lambda ](t) = {{\tilde{I}}}[\lambda ^2](t)\) for \(t\ge t_0\).
Instead of finding \(\lambda \) such that \(I[\lambda ]=0\) for \(t>t_0\) we are going to construct \(\mu \) such that
for some \(\sigma >0\).
Let \(\mu ^* = (\lambda ^*)^2\) where \(\lambda ^*\) is defined in (7.7). In a first step we will find \(\mu _1\) so that
We will look for \(\mu _1\) with \(\Vert \mu _1\Vert _{*,\gamma ,m}<\infty \) where, for a function \(\mu _1 \in C^1([\frac{t_0}{2}, \infty ))\) with \(\lim _{t\rightarrow \infty } \mu _1(t)=0\) we define
Equation (7.21) takes the form
where
and \(F_1\) is an operator with the following properties:
for \({\tilde{\mu }}_j\) satisfying \(\Vert {\tilde{\mu }}_j\Vert _{*,\gamma ,m} \le 1\), with \(0<\gamma <2\), \(m\in {\mathbb {R}}\), where \(\Vert \ \Vert _{\gamma ,m}\) is defined in (7.6). From (7.20) we find
Now we apply the contraction mapping principle to the Eq. (7.22) written in the form
where
We directly check that
Let X be the space \(X = \{ \mu _1 \in C^1([ \frac{t_0}{2},\infty ) ) \, | \, \lim _{t\rightarrow \infty } \mu _1(t) = 0 \}\) with the norm \(\Vert \mu _1 \Vert _X = \Vert \mu _1 \Vert _{*,1,3-\varepsilon }\), where \(0<\varepsilon <1\). It follows that if \(\vartheta <\frac{1}{2}\) the equation (7.25) has a unique solution \(\mu _1 \) in the ball \({{\overline{B}}}_1(0)\) of X.
Therefore we have found \(\mu _1\) with \(\Vert \mu _1 \Vert _{*,1,3-\varepsilon } \le 1\) so that \(\mu = \mu ^* + \mu _1\) satisfies
To estimate this remainder we then need a bound for \(\ddot{\mu }\). Differentiating with respect to t in the decompositions used in Lemmas 7.1, 7.4, 7.5 we obtain
Differentiating in t equation (7.25) and using the contraction mapping principle we get that for any \(\varepsilon >0\) small
Using this we find that the remainder (7.26) has the estimate
where \(\mu = \mu ^* + \mu _1\).
Next we introduce another correction \(\mu _2\) to improve the decay of the remainder. We consider \(\mu = \mu ^* + \mu _1 + \mu _2\) and we consider the following equation for \(\mu _2\):
Similarly as before, this equation can be written as
where \(F_2\) satisfies the same estimate (7.23) (7.24), and \(e_2\) has the estimate
Using again the contraction mapping principle we find a solution \(\mu _2\) of (7.27) with \( \Vert \mu _2\Vert _{*,1+\vartheta -\varepsilon ,1} \le 1\). Then for \(\mu = \mu ^* + \mu _1 + \mu _2\)
To estimate this remainder we need the following bound for \(\ddot{\mu }_2\)
which is obtained from an estimate for \(\dot{e}_2\), differentiating with respect to t equation (7.27). The estimate for \(\dot{e}_2\) is obtained from an analogous estimate for \(\frac{d^3 \mu _1}{dt^3}\).
From (7.28) we find
where we recall that \(0<\vartheta <\frac{1}{2}\) is arbitrary.
Thus letting \(\lambda _0 = \sqrt{\mu }\), \(\mu = \mu ^* + \mu _1 + \mu _2\) we obtain
Choosing \(\vartheta >\frac{1}{4}\) and \(\varepsilon >0\) small, we obtain the properties stated in Proposition 5.1. \(\square \)
8 Inner Linear Theory
In this section we consider the problem
that appears in the inner equations (5.48) and (5.49), where, we recall
Slightly more general than the operator B defined in (5.47) we will consider
where \([\phi ]_{rad}\) is the radial part of \(\phi \) (defined in (5.46)) and \(\phi _1 = \phi - [\phi ]_{rad}\), and where \(\chi _0\) is the smooth cut-off function defined in (2.5). In the sequel we will keep the same notation for B.
In what follows we will analyze the linear initial value problem (8.1) where we assume that the functions \(\lambda (t)\), \(\zeta _i(t)\) are continuous, \(t_0>1\) and that for some positive numbers c, C we have
We change the time variable into
where \(\tau _0 = t_0 \log t_0\). Then
for some \({{\tilde{c}}}_1,{{\tilde{c}}}_2>0\). Identifying \(\phi (y,t)\) and h(y, t) with \(\phi (y,\tau )\) and \(h(y,\tau )\) we rewrite (8.1) as
We consider problem (8.3) for functions \(h(y,\tau )\) that have fast decay in space. More precisely, we assume that for all \(T>0\) there is \(C_T\) such that
In this case, by a solution \(\phi (y,\tau )\) of (8.3) we understand a continuous function \(\phi (y,\tau )\), of class \(C^1\) in y, such that for any \(T>\tau _0\) there exists a \(C_T>0\) with
and satisfies the integral equation
where \((-\Delta )^{-1}\phi \) is defined in (8.2) and \(G(y,\tau )\) is the two-dimensional heat kernel,
From the formula
we see that if \(|\phi (y)|\le \frac{C}{1+|y|^6}\) then
Using this estimate, existence and uniqueness of a solution of (8.5) satisfying (8.4) are standard. For a short time \(T>\tau _0\) this is established by a contraction mapping argument in an appropriate \(L^\infty \)-weighted space. Then a direct linear continuation procedure applies.
A first natural condition to impose on h in (8.3) is that
in order to achieve that the solution has also zero mass at all times.
We want to find solutions to (8.3) that have fast decay in space and time. For this we need to assume fast space-time decay of the right hand side, which we do by working with the following class of norms.
Given positive numbers \(\nu \), p, \(\varepsilon \) and \(m\in {\mathbb {R}}\), we let \(\Vert h\Vert _{\nu ,m,p,\varepsilon }\) denote the least \(K\geqq 0\) such that for all \(\tau >\tau _0\) and for all \(y\in {\mathbb {R}}^2\)
This is similar to the norm introduced in (6.2) but defined using \(\tau \) instead of t. We will give the results in Sects. 9–12 using the norm (8.6).
Still, fast decay of the right hand side doesn’t imply fast decay of the solution. For example, consider Eq. (8.1) without the operator B and without the \(\mu _j\), that is,
and suppose that h has compact support in space and time, and that \(\phi \) has sufficient space-time decay. Then, multiplying (8.7) by \(|y|^2\) and integrating in \({\mathbb {R}}^2 \times (\tau _0,\infty ) \) gives
because if \(\phi \) is a regular function with fast decay, then
see Remark 9.2 below. It is then necessary to impose a condition on h, or to adjust a parameter in the problem in order to get a fast decay of the solution. We develop here the theory by adjusting the parameter \(c_1\) in the equation below
where \({{\tilde{Z}}}_0\) is defined in (6.4).
Proposition 8.1
Let \(\sigma >0\), \(\varepsilon >0\) with \(\sigma +\varepsilon <2\) and \(1<\nu <\frac{7}{4}\). Let \(0<q<1\). Then there exists a number \(C>0\) such that for \(t_0\) sufficiently large and all radially symmetric \(h=h(|y|,\tau )\) with \(\Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
there exists \(c_1 \in {\mathbb {R}}\) and solution \(\phi (y,\tau ) = {\mathcal {T}}^{i,2}_{{{\textbf {p}}}} [h]\) of problem (8.8) that defines a linear operator of h and satisfies the estimate
Moreover \(c_1\) is a linear operator of h and
We have stated this result only in the radial setting, because this is what is needed, but there is a version of it in the non-radial case.
The next result is for the problem
and holds without the radial symmetry assumption.
Proposition 8.2
Let \(0<\sigma <1\), \(\varepsilon >0\) with \(\sigma +\varepsilon <\frac{3}{2}\) and \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{5}{4})\). Let \(0<q<1\). Then there is C such that for \(\tau _0\) large the following holds. Suppose that h satisfies \(\Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
Then there exists a solution \(\phi (y,\tau )\), \(\mu _j\) of problem (8.9) that defines a linear operator of h and satisfies
The parameters \(\mu _j\) satisfy
where \({\tilde{\mu }}_j\) are linear functions of h with
We denote this solution by \(\phi = {\mathcal {T}}^{i,1}_{{{\textbf {p}}}} [h]\).
Propositions 6.1 and 6.2 given in Sect. 6 are direct corollaries of Propositions 8.1 and 8.2. The only changes are due to the change in the time variable, because \(\tau \sim t \log t\), and the fact that the norms for the solutions in Propositions 6.1 and 6.2 include a gradient term. The estimate for the gradient follows from the weighted \(L^\infty \) estimate, scaling and standard parabolic estimates.
The proofs of Propositions 8.1 and 8.2 are contained in Sects. 9–12. They are based on an energy inequality obtained by multiplying the equation by a suitable test function, and using an inequality for a quadratic form. Section 9 contains some preliminaries on this quadratic form.
In Proposition 10.1, we obtain an additive decomposition of the solution \(\phi (y,\tau )\) of (8.8) into a part with a relatively slow space decay that loses \(\tau ^{1/2}\) with respect to the time decay of the right hand side, and a term along \(Z_0(y)\) that loses an entire power of \(\tau \). This is the key element for the proof of Proposition 8.1 in Sect. 10 (p.80).
Then the proof of Proposition 8.2 in the radial case uses Proposition 10.1 after formally applying the operator \(L^{-1}\) to the original equation and performing a concentration procedure that improves the space decay of the resulting error. This is done on Sect. 11, and we give there a proof of Proposition 8.2 in the case of radial functions.
The proof of Proposition 8.2 in the general case is in Sect. 12 (p.96).. The idea is that the decomposition obtained in Proposition 10.1 for solutions with no radial mode does not contain the term along \(Z_0\), which allows us to obtain a much better estimate.
9 Preliminaries for the Linear Theory
A central ingredient in obtaining good estimates for the linearized parabolic operator associated to the inner problem is the analysis of the quadratic form
This quadratic form arises when considering the linearized Keller–Segel problem (8.1). Indeed, \(L[\phi ] = \nabla \cdot ( U \nabla g) \) and it is natural to test the Eq. (8.1) with g, since
But from the time derivative we get \(\lambda ^2 \int _{{\mathbb {R}}^2} \partial _t \phi g \), which leads to (9.1).
We observe that g has degeneracy directions. Indeed, if \(\psi = (-\Delta )^{-1} \phi \) then
The operator \(\Delta \psi + U(y) \psi \) is classical. It corresponds to linearizing the Liouville equation
around the solution \(\Gamma _0 = \log U\). It is well known that the bounded kernel of this linearization is spanned by the generators of rigid motions, namely dilation and translations of the equation, which are precisely the functions \(z_0,z_1,z_2\) defined by
Note that g is precisely annihilated at the linear combinations of these functions. In the rest of this section we will state and prove several estimates that take into account this issue, which will be crucial later on.
The quadratic form (9.1) can be naturally transformed into a similar one in \(S^2\) by stereographic projection \(\Pi :S^2\setminus \{(0,0,1)\}\rightarrow {\mathbb {R}}^2\)
For \(\varphi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) we write
Then we have the following formulas:
9.1 The Liouville equation
Here we consider the linearized Liouville equation
The stereographic projection transforms the linearized Liouville equation (9.3) into
in \(S^2{\setminus }\{P\}\), \(P = (0,0,1)\), where \({\tilde{\psi }} = \psi \circ \Pi \), \({{\tilde{h}}} = (U^{-1} h) \circ \Pi \).
The functions in (9.2) are transformed through the stereographic projection into constant multiples of the coordinate functions
By standard elliptic theory, if \({{\tilde{h}}} \in L^p(S^2)\), \(p>2\), then exists a solution \({\tilde{\psi }}_0 \in W^{2,p}(S^2)\) to (9.4) in \(S^2\) if and only if \({{\tilde{h}}}\) satisfies
This solution is unique if we normalize it such that
and then satisfies the estimate
where \(\alpha = 1- \frac{2}{p}\). By subtracting off a suitable linear combination of the functions \({{\tilde{z}}}_j\), \(j=0,1,2\) we obtain the unique solution \({\tilde{\psi }}_1\) to (9.4) in \(S^2\) satisfying
For this solution we also have the estimate
Lemma 9.1
Let \(0<\sigma <1\). Then there is C such that if \(\psi \) satisfies (9.3) and \(\psi (y)\rightarrow 0\) as \(|y|\rightarrow \infty \) with h satisfying \(\Vert (1+|y|)^{3+\sigma }h\Vert _{L^\infty ({\mathbb {R}}^2)}<+\infty \) and
then
Remark 9.1
Let \(h:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) satisfy \(\Vert (1+|y|)^{2+\sigma }h\Vert _{L^\infty ({\mathbb {R}}^2)}<+\infty \) where \(0<\sigma <1\). If
then
If \(h:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) satisfy \(\Vert (1+|y|)^{3+\sigma }h\Vert _{L^\infty ({\mathbb {R}}^2)}<+\infty \) where \(0<\sigma <1\) and in addition to mass zero we have
then
The first claim is standard. For the second, write
and estimate the integral after splitting it into the regions \(|y|<\frac{|x|}{2}\) and its complement.
Proof of Lemma 9.1
We claim that \(\psi = (-\Delta )^{-1} (U\psi +h)\). Indeed the function \(\psi - (-\Delta )^{-1} (U\psi +h)\) is harmonic in \({\mathbb {R}}^2\) and decays to 0 at infinity, and therefore it is equal to 0. The assumptions (9.7) and Remark 9.1 imply that
Let \({\tilde{\psi }} = \psi \circ \Pi \), so that it satisfies (9.4) in \(S^2\setminus \{P\}\) with \({{\tilde{h}}} = (U^{-1} h)\circ \Pi \). Note that \({{\tilde{h}}} \in L^p(S^2)\) for some \(p>2\). More precisely,
with \(p<\frac{2}{1-\sigma }\). The singularity at P is removable and thus \({\tilde{\psi }}\) satisfies (9.4) in \(S^2\). By elliptic regularity \({\tilde{\psi }} \in C^{1,\alpha }(S^2)\) for some \(\alpha >0\). Since \(\psi \) decays at infinity, \({\tilde{\psi }}(P)=0\). By (9.8) we have also \(\nabla _{S^2} {\tilde{\psi }}(P)= 0\).
We let \({\tilde{\psi }}_1\) denote the solution to (9.4) satisfying (9.5). The solution to (9.4) in \(S^2\) satisfying (9.5) is unique, so that we have \({\tilde{\psi }} = {\tilde{\psi }}_1\) and by estimate (9.6), (9.9) and (9.8) we obtain
\(\square \)
9.2 A quadratic form
Here we discuss properties of the quadratic form (9.1). For this we consider a function \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) with sufficient decay, in the form
with \(0<\sigma <1\), and zero mass,
We recall g defined in (9.1) \(g = \frac{\phi }{U} - (-\Delta )^{-1}\phi \), and use the notation
so that
We next introduce a normalized version of g, namely \(g^\perp \) defined by
where \(a \in {\mathbb {R}}\) is chosen so that
As shown in Lemma 9.3 below, the quadratic form \(\int _{{\mathbb {R}}^2} \phi g\) is equivalent to \(\int _{{\mathbb {R}}^2} U(g^\perp )^2 \).
It will be convenient to work with functions \(\phi ^\perp \), \(\psi ^\perp \), which are analogues of \(\phi \), \(\psi \) but associated to \(g^\perp \). In particular, we want a choice of \(\psi ^\perp \) such that
Let \(\psi _0=1+\frac{1}{2}z_0\), where \(z_0\) is defined in (9.2), and observe that
Then \(\psi ^\perp \) defined by
indeed satisfies (9.12).
Define
and obtain the relations
We note that \(\phi - \phi ^\perp = \frac{a}{2} U z_0\) is a constant times \(Z_0 = U z_0\), which is in the kernel of the operator L.
Lemma 9.2
If \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) satisfies (9.10) and (9.11), then
where \(z_j\) are the functions defined in (9.2).
Proof
By the definition of \(\psi \) and from (9.10), (9.11) we have
and hence also
We multiply (9.12) by \(z_j\), integrate in the ball \(B_R(0)\) and let \(R\rightarrow \infty \). Since \(z_j\) is in the kernel of \(\Delta +U\) we just have to check that
where \(\nu \) is the exterior normal vector to \(\partial B_R\). This follows from (9.13), and the explicit bounds
\(\square \)
A consequence of the previous lemma is the following.
Remark 9.2
Suppose that \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) satisfies (9.10) and (9.11). Then
Indeed, integrating on \(B_R\), with the notation \(g = \frac{\phi }{U}-(-\Delta )^{-1}\phi \),
By (9.10) and (9.11), \(g(y) = O ( |y|^{2-\sigma } )\), \(\nabla g(y) = O ( |y|^{1-\sigma } )\) as \(|y|\rightarrow \infty \). Therefore the boundary terms tend to 0 as \(R\rightarrow \infty \), and we get
by Lemma 9.2.
Lemma 9.3
There are constants \(c_1>0\), \(c_2>0\) such that if \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) satisfies
and (9.11), then
Proof
By Lemma 9.2
Let \({{\tilde{g}}}^\perp = g^\perp \circ \Pi \), \({\tilde{\psi }}^\perp = \psi ^\perp \circ \Pi \) and write (9.12) as
We also get
Multiplying (9.15) by \({\tilde{\psi }}^\perp \) we find that
and hence
We recall that the eigenvalues of \(-\Delta \) on \(S^2\) are given by \( \{ k ( k+1) \ | \ k \ge 0 \} \). The eigenvalue 0 has a constant eigenfunction and the eigenvalue 2 has eigenspace spanned by the coordinate functions \(\pi _i(x_1,x_2,x_3) = x_i\), for \((x_1,x_2,x_3) \in S^2\) and \(i=1,2,3\). Let \((\lambda _j)_{j\ge 0}\) denote all eigenvalues, repeated according to multiplicity, with \(\lambda _0 = 0\), \(\lambda _1=\lambda _2=\lambda _3=2\), and let \((e_j)_{j\ge 0}\) denote the corresponding eigenfunctions so that they form an orthonormal system in \(L^2(S^2)\), and \(e_1,e_2,e_3\) are multiples of the coordinate functions \(\pi _1,\pi _2,\pi _3\). We decompose \({\tilde{\psi }}\) and \({{\tilde{g}}}\) to get that
where
Then
Equation (9.15) gives us that
and then
By Lemma 9.2\({{\tilde{g}}}^\perp _1 = {{\tilde{g}}}^\perp _2 = {{\tilde{g}}}^\perp _3 = 0\). Therefore
and
This proves (9.14). \(\square \)
Lemma 9.4
There exist positive constants \(c_1\), \(c_2\) such that if \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) is radially symmetric and satisfies \((1+|y|)^{3+\sigma } \phi \in L^\infty ({\mathbb {R}}^2)\) with \(0<\sigma <1\), and
then
Proof
Using the same notation as in the proof of Lemma 9.3, we have
As in the previous proof, \({{\tilde{g}}}^\perp _j=0\) for \(j=0,1,2,3\). Using (9.17) we get
This formula already gives
We observe that \({\tilde{\psi }}^\perp _1={\tilde{\psi }}^\perp _2=0\) by radial symmetry. We also have \({\tilde{\psi }}^\perp _0=0\), by (9.17). Let
and note that it satisfies
By (9.17),
and from elliptic estimates
for any \(0<\alpha <1\). Since \((1+|y|)^{3+\sigma }\phi \in L^\infty ( {\mathbb {R}}^2 )\) and \(\phi \) has total mass 0, we have \((1+|y|)^{1+\sigma }\psi \in L^\infty ( {\mathbb {R}}^2 )\) (here the functions are radial) and also \((1+|y|)^{1+\sigma }\psi ^\perp \in L^\infty ( {\mathbb {R}}^2 )\). It follows that \({\tilde{\psi }}^\perp (P)\) = 0 where \(P= (0,0,1)\). Since \({\tilde{\psi }}^\perp \) and \({\hat{\psi }}\) differ by a constant times \(\pi _3\) we have
where \(\pi _3(x_1,x_2,x_3)=x_3\). This implies, by (9.21),
This proves the other inequality in (9.19) and (9.20). \(\square \)
Lemma 9.5
Suppose that \(\phi = \phi (y,t)\), \(y\in {\mathbb {R}}^2\), \(t>0\) is a function satisfying
with \(0<\sigma <1\),
and that \(\phi \) is differentiable with respect to t and \(\phi _t\) satisfies also
Then
where for each t, g(y, t) is defined as
and \(c(t)\in {\mathbb {R}}\) is chosen so that
Proof
Using the notation of the previous lemma, we have
We have
And differentiating in t we get
Multiplying by \({{\tilde{g}}}\) and integrating we find that
Thus
Decompose as in (9.16) and find that
But from (9.22)
We note that \({{\tilde{g}}}_j=0\) for \(j=0,1,2,3\). Indeed, this is true for \(j=0\) by the assumption \(\int _{{\mathbb {R}}^2} g U=0\). By Lemma 9.2 this is true also for \(j=1,2,3\). Then
and the desired conclusion follows from (9.18). \(\square \)
9.3 A Poincaré inequality
Lemma 9.6
Let \(B_R(0)\subset {\mathbb {R}}^2\) be the open ball centered at 0 of radius R. There exists \(C>0\) such that, for any \(R >0\) large and any \(g\in H^1(B_R)\) with \(\int _{B_R} g\,U\,dx =0\) we have
Proof
Using a Fourier decomposition we only need to consider the radial case, that is, we claim that if g(r) satisfies
then there is C such that for all R large
Let \(0<\delta <1\) to be fixed later on. From (9.23) we have
But
Therefore
By the Cauchy-Schwarz inequality
Hence
We compute now
Using (9.24) and the Cauchy-Schwartz inequality we get
But \(\frac{1}{AR^2 r} \le \frac{1}{2}\frac{r}{(1+r^2)^2}\) for \(r\in [\delta ,R]\) if \(A = 4 (1+\frac{1}{\delta ^2})\) and \(R\ge 1\). Choosing \(A = 4 (1+\frac{1}{\delta ^2})\) and \(R\ge 2\) we have
With \(\delta >0\) still to be chosen we get from (9.24) for \(0<x<\delta \)
Integrating we get
Using the condition (9.23) we obtain
Then
Using this combined with (9.26) we get
Taking \(\delta =\frac{1}{2}\) (this fixes A) gives
Combining this with (9.25) we get
\(\square \)
10 Linear Theory: A Decomposition
Here we consider
The results of this section are going to be used later only in the case of radial functions, so we make this assumption here. We write in the rest of this section \(\phi = \phi ( y,\tau ) = \phi (\rho ,\tau )\), where \(y\in {\mathbb {R}}^2\), \(\rho = |y|\).
The operator B is assumed to be one of the following two:
or
where
for some constants \(\zeta _0>0\), \(0<\sigma _0<1\).
We assume that \(\Vert h\Vert _{**}<\infty \) where
where \(\nu >1\), \(\varepsilon >0\), \(\sigma >0\), \(m\in {\mathbb {R}}\). This is the same norm as in (8.6).
We also assume that h has zero mass
and the same for the initial condition
It follows from the equation (10.1), (10.4), and (10.5) that the solution \(\phi \) to (10.1) defined in §8 satisfies
We recall the decomposition of \(\phi \) introduced in §9.2. Given \(\phi :{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) with sufficient decay and mass zero, we let \(g = \frac{\phi }{U}- (-\Delta ^{-1})\phi \), and define a so that \(\int _{{\mathbb {R}}^2} ( g + a ) U dy= 0\). Then define \(g^\perp = g + a \), \(\psi ^\perp = \psi - a( 1+\frac{1}{2}z_0)\), and
Actually a is directly computed by
In the time dependent situation \(a = a(\tau )\) and all functions depend on \(y\in {\mathbb {R}}^2\) and \(\tau \).
A difficulty to obtain estimates is the presence of a kernel in the linear operator if \(B=0\), since \(Z_0\) satisfies \(L[Z_0]=0\). It can be proved that the solution \(\phi \) of (10.1) with zero initial condition and \(\Vert h\Vert _{**}<\infty \) has the bound
and probably this estimate cannot be improved much. Also \(\phi \) has a some decay at spatial infinity and in particular it has finite second moment
Therefore \(Z_0\) doesn’t describe well the class of solution we want to consider, even for the case \(B=0\), in which \(\zeta (\tau ) \equiv 0\).
A better candidate to describe the solutions \(\phi \) of (10.1) with zero initial condition and \(\Vert h\Vert _{**}<\infty \) is obtained by considering the initial value problem
where \( {{\tilde{Z}}}_0\) is defined in (6.4). Note that since \(Z_0\) has mass zero and decays like \(1/\rho ^4\) we have \( m_{Z_0} = O ( \frac{1}{\tau _0} ) \).
We will then consider the problem
for radial functions \(\phi \), h, \(\phi _0\), where \(c_1\in {\mathbb {R}}\) is a parameter. We assume that \(\Vert h\Vert _{**}<\infty \).
Proposition 10.1
Let us assume that \(1<\nu <\frac{7}{4}\). Then there is \(C>0\) such that for any \(\tau _0\) sufficiently large the following holds. Suppose that \(\Vert h\Vert _{**}<\infty \) is radially symmetric and satisfies the zero mass condition (10.4). Then there exists \(c_1\) such that the solution \(\phi = \phi ^\perp + \frac{a}{2}Z_0\) of (10.9) satisfies
where \(R(\tau )>0\) is defined by
where \(0< q<1\), and
Moreover \(c_1\) is a linear function of h and satisfies
We always decompose \(\phi \) as in (10.6):
and write
Let us denote
The strategy for the proof of Proposition 10.1 is contained in the following lemmas. The first one is an a-priori estimate for the solution, assuming that \(a(T_2)=0\) for some \(T_2\).
Lemma 10.1
There is C such that for \(\tau _0\) large the following holds. Suppose that \(\Vert h\Vert _{**}<\infty \) is radially symmetric and satisfies the zero mass condition (10.4) and consider (10.9). Let \(\phi ^\perp \), a be the decomposition (10.6). Suppose that for some \(c_1\in {\mathbb {R}}\) there is \(T_2>\tau _0\) is such that
Then
The constant C is independent of \(T_2\) and \(c_1\).
There is a variant of the previous lemma, where the hypothesis \(a(T_2)=0\) is replaced by an assumption about its time decay.
Lemma 10.2
There is C such that for \(\tau _0\) large the following holds. Suppose that \(\Vert h\Vert _{**}<\infty \) is radially symmetric and satisfies the zero mass condition (10.4) and consider (10.9). Let \(\phi ^\perp \), a be the decomposition (10.6). Suppose that for some \(c_1\in {\mathbb {R}}\),
Then
Lemma 10.3
Let \(Z_B\) be the solution to (10.8) and write it as \(Z_B = Z_B^\perp + \frac{a_Z}{2}Z_0\) according to the decomposition (10.6). Then \(a_Z(\tau )\not =0\) for all \(\tau \ge \tau _0\).
Lemma 10.4
There is C such that for \(\tau _0\) large the following holds. Suppose that \(\Vert h\Vert _{**}<\infty \) is radially symmetric and satisfies the zero mass condition (10.4). Then there is a unique \(c_1 \in {\mathbb {R}}\) such that the solution \(\phi = \phi ^\perp + \frac{a}{2}Z_0\) of (10.9) (as in (10.6)) satisfies (10.17), (10.18) and (10.19).
In the first results we do some computations and obtain some estimates, which are used as technical steps in the main argument.
The next lemma is a calculation to help us deal with the term B when we multiply the equation by a suitable test function. It holds for operators more general than B as in (10.2) and (10.3). Let
with \(\zeta _1(\tau )\), \(\zeta _2(\tau )\) satisfying
Lemma 10.5
We have
Proof
We have
By Lemma 9.3 and the hypothesis (10.20) we have
Let us write
We claim that
Indeed, we write
But
and so
The second term in (10.24) is:
We estimate the first term above
by (9.20). To estimate \(\int _{{\mathbb {R}}^2} U (y \cdot \nabla \psi ^\perp ) g^\perp (y)dy \) we write it using radial symmetry:
We use that \(\psi ^\perp \) satisfies
Then, by the variations of parameters formula, since that \(\int _{{\mathbb {R}}^2} U g^\perp z_0 dy=0\), we have
where \({{\bar{z}}}_0\) is a second linear independent function in the kernel of \(\Delta +U\) satisfying
We then compute
where
We directly check that
From this we get that
Combining (10.24), (10.25), (10.26), (10.27) we obtain (10.23).
Next we claim that
Indeed, write
where \(z_0\) is defined in (9.2) and satisfies the linearized Liouville equation \(\Delta z_0 + U z_0=0\). We have used here that \(Z_0 = U z_0\). So
But \(\int _{{\mathbb {R}}^2} Z_0 g^\perp dy=\int _{{\mathbb {R}}^2} U z_0 g^\perp dy=0\) by Lemma 9.2, and \(|y Z_0 - 2 \nabla z_0| \le \frac{C}{|y|^4}\), so
This proves (10.28).
From (10.22), (10.23) and (10.28) we conclude the validity of (10.21).
\(\square \)
In the next lemma we get an estimate for \(\int _{{\mathbb {R}}^2} \phi g^\perp \), but with right hand side that depends on the solution.
Lemma 10.6
We make the same assumptions of Proposition 10.1. Let f be given by (10.12), \(\omega \) be defined in (10.13) and let \(R:[\tau _0,\infty ) \rightarrow (0,\infty )\) be continuous. There is \(c>0\), \(\varepsilon >0\) and \(C>0\) such that for \(\tau _0\) sufficiently large, if
then
for some constant \(c>0\).
Proof
Equation (10.9) can be written in the form
We multiply this equation by \(g^\perp \) and integrate on \({\mathbb {R}}^2\), using Lemma 9.5:
Let \(H = (-\Delta )^{-1}h\), and observe that, since h is radial and \(\int _{{\mathbb {R}}^2} h dy = 0\),
It follows that
This combined with (10.30) gives
We use the inequality in Lemma 9.6 to get
for some \(c>0\), where
From
we get
so, using (10.32),
for a new \(c>0\). This implies
Using that \(g^\perp = g + a\) we get
But
and this implies
so
This combined with (10.33) gives
We use this together with (10.31) to obtain (for a new \(c>0\))
We obtain from Lemma 10.5 and the assumption (10.29) that
Taking \(\varepsilon >0\) small, and combining (10.34) and (10.35) we get
By Lemma 9.3 we obtain
for some constant \(c>0\), which is the desired conclusion. \(\square \)
The next lemma provides a pointwise estimate for \(g = \frac{\phi }{U}- (-\Delta ^{-1})\phi \) assuming a certain bound for \(\Vert U^{1/2} g \Vert _{L^2}\).
Lemma 10.7
Assume \(\nu >0\). Let \(\phi \) be the solution to (10.9) as in §8. Suppose that \(\tau _1\ge \tau _0\) and
where \(K_1 \ge 0 \) and
where \(\mu \in {\mathbb {R}}\). Then
Proof
We define
and obtain from (10.1) the equation
where we regard \(\psi [g_0]\) as the operator that maps \(g_0\) to the unique radial solution to
We note that this problem has indeed a solution since \(\int _{{\mathbb {R}}^2} g_0 z_0 dy = 0\) by Lemma 9.2, which is unique by imposing \(\psi (\rho ,\tau ) \rightarrow 0\) as \( \rho \rightarrow \infty \) in the radial setting. This solution is given by the variations of parameters formula
where \({{\bar{z}}}_0\) is a second linear independent function in the kernel of \(\Delta +U\) satisfying \( |{{\bar{z}}}_0(\rho ) | \le C( |\log \rho | + 1) \).
We compute
and hence
where
We write (10.37) as
where
Note that since we are working with radial functions, we can integrate (10.39) explicitly and obtain
We claim that for any \(y \in {\mathbb {R}}^2\):
Indeed, let us start with
which follows from (10.42) and Hölder’s inequality
Let us write \(\psi =\psi [g_0]\) and \({\tilde{\psi }} = \psi \circ \Pi \), where \(\Pi \) is the stereographic projection. Writing (10.38) in \(S^2\) and using standard \(L^p\) theory we find that for any \(p>2\)
which implies
Let \(y\in {\mathbb {R}}^2 \). From (10.36) we see that
and from (10.36) and (10.44) we have
Similarly, inequalities (10.45) and (10.36) imply
Let’s estimate
Note that \(\psi = (-\Delta )^{-1}\phi = (-\Delta )^{-1}(g_0 + U \psi ) \). But we can estimate \(\psi \) from
Then (10.36) yields
and so
Concerning the term \((-\Delta )^{-1} ( y \cdot \nabla ( g_0 + U \psi ))\), we notice that if we let \(w = g_0 + U \psi \), then \(\int _{{\mathbb {R}}^2} y \cdot \nabla w = 0\), and
Using (10.36) and (10.49) we get
From this and (10.50) we find that
Finally the estimates
are directly obtained.
Combining (10.46), (10.47), (10.51), and (10.52) we deduce (10.43).
From equation (10.40), the estimate (10.43), standard parabolic \(L^p\) estimates restricted to \(B_1(y)\times ( \max (\tau -1,\tau _0),\tau )\) and embedding into Hölder spaces, we deduce that
This is the desired conclusion. We also get from (10.53):
\(\square \)
In some of the proofs below the following barrier will be useful. Consider the equation
where \(\Delta _{{\mathbb {R}}^6}\) is the laplacian in \({\mathbb {R}}^6\). Suppose that h has the estimate
for some \(\gamma ,b\in {\mathbb {R}}\).
If \(\gamma <3\) and \(\gamma <\frac{b}{2}\) then there is a barrier satisfying
Indeed, we can consider all functions to be radial and write \(\rho = |y|\), \(y\in {\mathbb {R}}^6\). Let
Then
Let \(g_1(\zeta ) = \frac{1}{(1+\zeta ^2)^{b/2}}\). Since \(\gamma <\frac{b}{2}\) we have
for some \(c, M>0\). Let \(g_0(\zeta ) = e^{-\frac{\zeta ^2}{4}}\) be the Gaussian kernel, which satisfies
Let \(g = C_1 g_0 + g_1\). Since \(\gamma <3\), we can find \(C_1\) large so that
Then \({\bar{\phi }}\) defined by (10.56) with \(g = C_1 g_0 + g_1\) is a supersolution to (10.55).
In the next lemma we improve the spatial decay of \(g = \frac{\phi }{U}- (-\Delta ^{-1})\phi \).
Lemma 10.8
Assume \(1<\nu <\frac{7}{4}\). Let \(\phi \) be the solution to (10.9) as in §8. Suppose that \(\tau _1\ge \tau _0\) and
where \(K_1\ge 0\) and
where \(\mu \in {\mathbb {R}}\). Then
Proof
We us the same notation as in Lemma 10.7 and consider (10.40) for \(g_0 = U g\) with \({{\tilde{h}}}\) defined in (10.41). We are going to use barriers to estimate \(g_0\).
We claim that \({{\tilde{h}}}\) satisfies
Indeed, from (10.53) and (10.54) we find that
To estimate \(B[U \psi [g_0]]\) we use (10.49) a similar estimate for \(\partial _\rho \psi \), and the assumptions on \(\zeta _1\), \(\zeta _2\) in (10.20), to obtain
This, (10.59) and (10.51) prove (10.58).
To get better spatial decay we construct a barrier and apply the maximum principle to equation (10.40) in \(({\mathbb {R}}^2 {\setminus } B_{R_0} (0) ) \times (\tau _0,\tau _1)\), where \(R_0\) is a fixed large constant. Several of constants C below depend on \(R_0\) but we will not keep track of the explicit dependence.
The linear operator for \(g_0\) in (10.40), acting on radial functions with \(\rho = |y|\), is given by
The main part outside of a ball \(B_{R_0}(0)\) with \(R_0\) big is given by \(\partial _\tau - \partial _{\rho \rho } - \frac{5}{\rho }\partial _\rho \).
By (10.58) we need to construct \({{\bar{g}}}_1\) such that
where
To construct \({{\bar{g}}}_1\), let \(0<\vartheta <1\), and let \({{\tilde{g}}}_1(\rho )\) be radial and solve
such that \( {{\tilde{g}}}_1(\rho ) (1+\rho ^{4-\vartheta })\) is bounded below and above by positive constants. Let
For appropriate \(\delta >0\), \(C_1\), and \(C_2\), the function \( {{\bar{g}}}_1(\rho ,\tau ) \) is a supersolution in \(({\mathbb {R}}^2 {\setminus } B_{R_0} (0) ) \times (\tau _0,\tau _1)\) for the right hand side \(h_1\). More precisely, writing \(M = R_0^{2-\vartheta } ( K_1 +\frac{\Vert h\Vert _{**}}{R(\tau _0)} + \frac{1}{f_1(\tau _0) } |c_1| )\), we have
because of Lemma 10.7, and
where
is the function g associated to \({{\tilde{Z}}}_0\) defined in (6.4). We note that \(|U g_{{{\tilde{Z}}}_0}(\rho )|\le C \frac{1}{1+\rho ^4}\) and is supported on \(\rho \le 2\sqrt{\tau _0}\). Here we are using that \(\nu <\frac{3}{2}+\frac{\vartheta }{2}\).
Using the maximum principle we get
The constant C here depends on \(R_0\), but \(R_0\) is fixed and we will not keep track of the dependence of C on \(R_0\).
By (10.42) and (10.48) we have
We can now repeat the argument with a new barrier. Consider \({{\tilde{g}}}_2(\rho )\) the radial solution to
where \(c_1\), \(c_2>0\). Let
For appropriate constants \(\delta \), \(C_1\), \(C_2\), and assuming that \(\nu <2-\frac{\vartheta }{2}\) we get a suitable supersolution and we obtain
This proves (10.57).
The restriction on \(\nu \) were \(\nu < \frac{3}{2}+\frac{\vartheta }{2}\) and \(\nu <2-\frac{\vartheta }{2}\). Choosing \(\vartheta =\frac{1}{4}\) we find that for \(\nu <\frac{7}{4}\) both barriers work. \(\square \)
The next result is a technical step used in several places.
Lemma 10.9
Let \(\phi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) be radial such that \(\int _{{\mathbb {R}}^2} \phi = 0\) and \(|\phi (y)|\le \frac{C}{(1+|y|)^{2+\sigma }}\) for some \(\sigma >0\). Let \(g = \frac{\phi }{U}-(-\Delta )^{-1} \phi \) and assume that \(\Vert g\Vert _{L^\infty } <\infty \). Then
Proof
Let \(\psi = (-\Delta )^{-1}\phi \). Since \(\psi \) satisfies
we have necessarily
We have the variations of parameters formula
From (10.62) we find
This and the formula \(\phi = U g + U \psi \) gives (10.61). \(\square \)
Next we give a proof of Proof of Lemma 10.1, but first we point some estimates of \({{\tilde{Z}}}_0\) defined in (6.4). Using the general decomposition (10.6), we write
By (10.7)
Hence \({{\tilde{Z}}}_0^\perp \) satisfies
where
Let \({{\tilde{g}}}_0=\frac{{{\tilde{Z}}}_0}{U}-(-\Delta _y)^{-1}{{\tilde{Z}}}_0\) and \({{\tilde{g}}}_0^\perp = {{\tilde{g}}}_0+{{\tilde{a}}}_0\). Note that since \(Z_0\) has mass zero and decays like \(1/\rho ^4\) we have \( m_{Z_0} = O ( \frac{1}{\tau _0} ) \). We claim that
Indeed, let us use the notation
so that
Let us write
where
Since \(\Delta z_0 + U z_0 = 0\) and \(\lim _{\rho \rightarrow \infty } z_0(\rho ) = -2\) we have \((-\Delta )^{-1} Z_0 = z_0+2\). Therefore
Since the mass of \({{\tilde{Z}}}_0\) is zero
But \(Z_0=Uz_0\) and the mass of h is zero, so
because, integrating by parts,
By direct computation
With this inequality we estimate
and
This proves (10.63).
Proof of Lemma 10.1
We let R be defined by (10.11). We multiply equation (10.9) by \(g^\perp \) and integrate in \({\mathbb {R}}^2\). Using Lemmas 10.6 and 9.3 we get
for some \(c>0\), where
Let us write
and note that
The following inequalities are valid for \(\tau _0<\tau <T_2\). From (10.64) we get
By Gronwall’s inequality and Lemma 9.3 we get
where
and we have used (10.63).
From (10.65) we find
Using Lemma 10.8 we get
where we have used that for \(\tau _0\) large, \( \frac{D(\tau _0)}{R(\tau _0)}< \frac{1}{f(\tau _0) R(\tau _0)^2} \).
We use this to estimate
which implies
We deduce that
Combining this inequality with (10.66) we obtain
and with (10.65) we get
Going back to (10.67) we find
Using Lemma 10.9 we also obtain
We multiply the equation satisfied by \(\phi \) (10.9) by \(|y|^2 \chi _0(\frac{y}{R})\), and integrate on \({\mathbb {R}}^2\)
where \(R' = \frac{d R}{d \tau }\).
We integrate (10.73) from \(\tau \) to \(T_2\), use the decomposition (10.6) and that \(a(T_2)=0\) to get
Analogously,
Integrating by parts
Let’s estimate, using (10.72)
The second term in (10.77) is even smaller, and we deduce that
From (10.72) we also get
Next we look at
We have \(\int _{{\mathbb {R}}^2} g Z_0 = 0\) by Lemma 9.2 and therefore, using (10.71), we find that
The remaining terms in (10.80) are estimated using (10.69) or (10.71) and we get
Therefore
Finally
From (10.74), (10.75), (10.76), (10.78), (10.79), (10.81), and (10.82) we get
Assuming \(\tau _0\) large, we deduce that
Note that \(a(\tau _0)\) and \(c_1\) are related. Indeed, the initial condition is \(\phi _0 = c_1 {{\tilde{Z}}}_0 = \phi _0^\perp + \frac{a(\tau _0)}{2} Z_0\) with
by (10.7). We note that \( \int _{{\mathbb {R}}^2} {{\tilde{Z}}}_0 \Gamma _0 = 16 \pi + O(\frac{\log \tau _0}{\tau _0})\). So by (10.84)
For \(\tau _0\) large, we deduce that
This proves (10.16). Replacing this in (10.84) we get
which proves (10.14). Combining (10.68), (10.85) and (10.86) we obtain (10.15).
Finally, we also obtain from (10.72)
\(\square \)
Proof of Lemma 10.2
The proof is a slight modification of the one of Lemma 10.1. Using the same notation as in that proof, integrating (10.73) from \(\tau \) to \(T_2>\tau \) yields
Similarly to (10.83) we obtain
The assumption \(\frac{a}{f R^2} \in L^\infty (\tau _0,\infty )\) implies that
Letting \(T_2\rightarrow \infty \) in (10.88) we obtain
Then the same argument as in Lemma 10.1 gives the estimates for a, \(\omega \) and \(c_1\). \(\square \)
Proof of Lemma 10.3
Here \(Z_B\) is the solution to (10.8). Assume to the contrary that there is some \(T_2>\tau _0\) such that
We follow the same computations as in the proof of Lemma 10.1 with \(h=0\) and \(c_1=1\). By the inequality (10.84) in the proof of Lemma 10.1
which implies
But by (10.7)
which contradicts (10.89). \(\square \)
Proof of Lemma 10.4
We let \(T_n \) be a sequence such that \(T_n\rightarrow \infty \) as \(n\rightarrow \infty \). Let \({\bar{\phi }}\) be the solution to (10.1) with initial condition equal to 0. This solution exists but for the moment we don’t have any control of its asymptotic behavior as \(\tau \rightarrow \infty \). Let \({\bar{\phi }}^\perp \), \({{\bar{a}}}(\tau )\) be the decomposition (10.6) of \({\bar{\phi }}\). Let \(Z_B^\perp \), \(a_Z(\tau )\) be the decomposition (10.6) of \(Z_B\). Using Lemma 10.3 there is \(c_n \in {\mathbb {R}}\) such that
Let us define
and let
be the decomposition (10.6) of \(\phi _n\). Then by Lemma 10.1 we have
Moreover, we also have the uniform estimate
for \(\tau \in [\tau _0,T_n]\) from (10.87).
By using standard parabolic estimates, passing to a subsequence we may assume that \(c_n\rightarrow c_1\) and \(\phi _n\rightarrow \phi \) locally uniformly in space-time, and that \(\phi \) is a solution of (10.9) for some \(c_1\) such that
Moreover \(\phi \) satisfies
and writing the decomposition (10.6) as \(\phi = \phi ^\perp + \frac{a}{2}Z_0\) we have
We also get
where \(\omega \) is defined in (10.13).
The uniqueness of \(c_1\) is a consequence of Lemma 10.2. \(\square \)
Proof of Proposition 10.1
We have already constructed \(\phi \) and \(c_1\) in Lemma 10.4, we have the uniqueness of \(\phi \) and the estimates for a and \(c_1\) in Lemma 10.2.
We only need to prove the estimate for \(\phi ^\perp \) stated in (10.10). By the construction of \(\phi \) in Lemma 10.4 and (10.70), (10.85) and (10.86), we get
We claim that from this inequality we have
The proof of this estimate is similar to that of (10.57) in Lemma 10.8.
Indeed, we define
and obtain the equation
Here the notation \(\psi [g_0^\perp ]\) is the one introduced in the proof of Lemma 10.7 in (10.38).
To get an estimate for the solution we need an estimate for \(a'(\tau )\). Since \(g^\perp = g + a\) and \(\int _{{\mathbb {R}}^2} U g^\perp =0\) we have
But integrating (10.37) we find
which gives the expression
We claim that
Indeed, we have
Then, by (10.90)
We also have, for the case of the operator (10.2),
But by construction and (10.69), (10.85) and (10.86), we get
so, using (10.93)
The last term is estimated similarly and we get (10.92).
Repeating the argument in of Lemma 10.7 we obtain from (10.90)
An argument similar to Lemma 10.9 gives
\(\square \)
We have an estimate for \(\phi ^\perp \) stronger than (10.10) under a stricter assumption on \(\nu \).
Lemma 10.10
Let us assume that \(1<\nu <\frac{3}{2}\). Under the same assumption of Proposition 10.1 let \(\phi = \phi ^\perp + \frac{a}{2}Z_0\) be the solution of (10.9). Then
Proof
We write (10.91) as
where
Then, similarly to (10.58), we have
Let
where \(-\Delta _6 {{\tilde{g}}}_3 = \frac{1}{1+\rho ^4}\) with \({{\tilde{g}}}_2(\rho )\rightarrow 0\) as \(\rho \rightarrow \infty \). If \(\nu <\frac{3}{2}\), for appropriate positive constants \(\delta \), \(A_1\), \(A_2\), and C, the function \(C \Vert h\Vert _{**} {{\bar{g}}}^\perp \) is supersolution to (10.94) in \(\{ (y,\tau ) | \tau> \tau _0, \ |y|> R_0 \}\). We deduce that
An argument similar to Lemma 10.9 gives
\(\square \)
Proof of Proposition 8.1
By Proposition 10.1 there is \(c_1\) such that the solution \(\phi \) to (10.9) has the properties stated in Proposition 10.1. We recall that by (10.87) \(\phi \) satisfies
We will construct a barrier to estimate \(\phi \) for \(|y|\ge R_0\), where \(R_0\) is a large constant. We consider the equation (10.9) in \({\mathbb {R}}^2 \setminus B_{R_0}(0)\) written in the form
where
Since \(\psi = ( -\Delta )^{-1}\phi \), from (10.95) we get
This gives
By (10.97) and the definition of the norm \( \Vert h\Vert _{**}\),
where we have used that \(\sigma +\varepsilon <2\). Let \({{\tilde{g}}}_2\) be defined by (10.60) and let
Then for suitable positive constants \(\delta \), \(A_1\), \(A_2\), and C, the function \(C (\log \tau _0)^{q-1} \Vert h\Vert _{**} {\bar{\phi }} \) is a supersolution to (10.96) in \(\{ (y,\tau ) | \tau> \tau _0, \ |y|> R_0 \}\). For this we need \(\nu <2\). Moreover \(|\phi (\rho ,\tau )| \le C {\bar{\phi }}(\rho ,\tau ) (\log \tau _0)^{q-1} \Vert h\Vert _{**}\) at \(\rho = R_0\) by (10.95). By the maximum principle
This gives the explicit bound
\(\square \)
We include here some results that will be useful later. Let
Lemma 10.11
The function \({{\hat{Z}}}_0\) satisfies
and is supported on \(\rho \le 2 \tau _0\).
Proof
Let \( \psi = (-\Delta )^{-1} {{\tilde{Z}}}_0\) and \(g = \frac{{{\tilde{Z}}}_0}{U} - \psi \). By (6.4) and using that \(Z_0 = U z_0\), \(z_0\) defined in (9.2),
where \(\chi (\rho ) = \chi _0(\frac{\rho }{\sqrt{\tau _0}})\). Note that \({{\tilde{Z}}}_0\) has mass zero and support in \(B_{2\sqrt{\tau _0}}\). It follows that \(\psi \) has also support contained in \(B_{2\sqrt{\tau _0}}\) and then g has support contained in \(B_{2\sqrt{\tau _0}}\). Therefore \({{\hat{Z}}}_0 = L[{{\tilde{Z}}}_0] = \nabla \cdot ( U \nabla g)\) has also support contained in \(B_{2\sqrt{\tau _0}}\).
To get an estimate for \({{\hat{Z}}}_0\) let us write
where
Since \(\Delta z_0 + U z_0 = 0\) and \(\lim _{\rho \rightarrow \infty } z_0(\rho ) = -2\) we have \((-\Delta )^{-1} Z_0 = z_0+2\). So
Hence
and so
Using radial symmetry and \(m_{Z_0} = O ( \frac{1}{\tau _0})\) we get
From this and (10.99) we get (10.98). \(\square \)
Consider the initial value problem
Lemma 10.12
Let \(0<\gamma <2\). Let \(1<\nu _0<\frac{7}{4}\)
and let \(R(\tau ) \) be as in (10.11). Then the solution \(\phi _1\) of (10.100) satisfies
Proof
A suitable modification in the proof of Proposition 10.1 gives the following result. Consider
Then there is \(C>0\) such that for any \(\tau _0\) sufficiently large the following holds. Suppose that \(\phi _0\) is a radial function with zero mass in \({\mathbb {R}}^2\), supported in \(B_{2\sqrt{\tau _0}}(0)\), and such that
Then there exists \(c_1\) such that the solution \(\phi \) of (10.101) satisfies
Moreover \(c_1\) is a linear function of \(\phi _0\) and satisfies
Let us apply this statement to \(\phi _0 = L[{{\tilde{Z}}}_0]\), which is radial, with mass zero, support in \(B_{2\sqrt{\tau _0}}(0)\), and satisfies
by Lemma 10.11. Then there exists \(c_1\) such that the solution \({\tilde{\phi }}\) to (10.101) with \(\phi _0 = L[{{\tilde{Z}}}_0]\) satisfies
We claim that \(c_1=0\). To prove this, we multiply (10.101) by \(|y|^2\) and integrate on \({\mathbb {R}}^2 \times (\tau _0,\infty )\). Let’s work with
The case of the operator (10.3) is similar. Then we get
because \(\int _{{\mathbb {R}}^2} L[\phi ]|y|^2dy=0\), see Remark 9.2. Integrating
because \(\int _{{\mathbb {R}}^2} L[{{\tilde{Z}}}_0] |y|^2 dy = 0\). Using the asymptotic expansion of \(\zeta \) one gets
But the bound (10.102) implies that
This only can happen if \(c_1=0\).
We deduce that \(\phi _1\) defined in (10.100) coincides with \({\tilde{\phi }}\), and then (10.102) holds for \(\phi _1\). \(\square \)
11 Linear Estimate with Second Moment (Radial)
We will prove in this section Proposition 8.2 in the radial case \(h(\rho ,\tau )\). In this case \(\mu _j\equiv 0\).
Proposition 11.1
Let \(0<\sigma <1\), \(\varepsilon >0\) with \(\sigma +\varepsilon <2\) and \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{5}{4})\). Let \(0<q<1\). Then there is C such that for \(\tau _0\) large the following holds. Suppose that h satisfies \(\Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
Then the solution \(\phi (y,\tau )\) of problem (8.9) satisfies
To describe the idea of the proof more easily let us consider for a moment the equation (8.9) without B:
The idea is to formally apply a suitable left inverse \(L^{-1}\) of L to (11.1) (to be defined later on in Lemma 11.1). If we call \(\Phi = L^{-1} \phi \), \(H = L^{-1}h\), then we would like to solve
In order to get good properties of H, in this step we have already used that h satisfies the second moment condition. At this point we would like to apply Proposition 10.1, which gives a decomposition
Note that \(\Phi ^\perp \) decays in time like \(1/\tau ^{\nu -1/2}\) and so \(\phi = L \Phi \) also decays in time like \(1/\tau ^{\nu -1/2}\), which is better than the estimate provided by Proposition 8.1. It turns out that H decays in space like \(1/\rho ^{4+\sigma }\) so we can’t apply directly Proposition 10.1 to (11.2). What we do is concentrate H by solving first a nicer problem. We write \(\Phi = \Phi _1 + \Phi _2\) where \(\Phi _1\) is asked to solve
where
Lemma 11.2 below deals with \(\Phi _1\). Then the problem for \(\Phi _2\) becomes
It turns out that the right hand side in this equation has better spatial decay and we can apply Proposition 10.1.
In the next lemmas we give some preliminary results, and the proof of Proposition 11.1 is given at the end of this section.
We define the inverse of L that we use. For \(h:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) define \(\Vert h\Vert _{\tau ,6+\sigma ,\varepsilon }\) as the smallest K such that
which depends on \(\tau \), treated as parameter here, \(\sigma \), and \(\varepsilon \).
Lemma 11.1
Let \(\sigma ,\varepsilon >0\). Let \(h=h(\rho )\) be radial and satisfy \(\Vert h\Vert _{\tau ,6+\sigma ,\varepsilon }<\infty \) and
Then there exists H radially symmetric such that \(L[H] = h\) in \({\mathbb {R}}^2\) and satisfies
Moreover, H defines a linear operator of h and satisfies
Proof
Write the equation \(L[H] = h\) as
where \(g = \frac{H}{U}- (-\Delta )^{-1} H\). We choose g as
Using that \(\int _{{\mathbb {R}}^2} h = 0\) we check that
Now we solve Liouville’s equation
Since \(\int _{{\mathbb {R}}^2} h |y|^2 dy=0\) we check that
Then we can use the variations of parameter formula, and get
Then define \(H = U ( g + \psi )\), which is the desired solution, and note that it satisfies (11.4). Property (11.5) follows from \(H = - \Delta \psi \) and the decay of \(\psi \). \(\square \)
To take into account the operator B we define
and compute
Indeed, write \(\Psi = (-\Delta )^{-1} \Phi \). Then
By direct computation
but \(-\Delta \Psi = \Phi \), and therefore
Applying \((-\Delta )^{-1}\) gives
Substituting this into (11.10) we obtain
Combining (11.7), (11.8), (11.9), (11.11) we find that
But
so that
Using that
we then obtain
Let’s consider the terms \(2 U \Phi - \nabla (\Lambda \Gamma _0)\cdot \nabla \Phi + \Lambda (U) \Phi \). Noting that \( \nabla (\Lambda \Gamma _0) = \nabla ( y \cdot \nabla \Gamma +2) = \nabla z_0\) and that \(Z_0 = 2 U + \Lambda (U)\), we can write
But \(\Delta z_0 + Z_0=0\), so
We can again write \(\nabla z_0 = \nabla ( y \cdot \nabla \Gamma _0)\) and using the radial symmetry of the functions \(\Gamma _0\), \(z_0\) and the notation \(\rho = |y|\)
Then
This proves (11.6).
Formula (11.6) leads us to consider the following equation for \(\Phi = L^{-1}[\phi ]\):
where
\(Z_0(y)= 2 U(y) + y \cdot \nabla _y U(y)\), and \({{\tilde{B}}}\) has the same form as B:
with \({\tilde{\zeta }}_1(\tau )\), \({\tilde{\zeta }}_2(\tau )\) satisfying
and \(\zeta _1\) satisfies the same restriction, that is, (10.20).
The next lemma allows us to reduce to an equation like (11.12) but with a right hand side with more spatial decay.
Lemma 11.2
Let \(\sigma >0\), \(\varepsilon >0\) and \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2})\). Let \(H(y,\tau )\) be radial in y and satisfy
and \(\Vert H\Vert _{\nu ,m,4+\sigma ,\varepsilon }<\infty \). Then there exists \(H_1\) and \(\Phi _1\) such that
Moreover \(\Phi _1\) and \(H_1\) are linear operators of H and satisfy
Proof
Write the operator L as
where \(L_0\) is defined in (11.3). Consider the problem
The idea is to formally apply \(L_0^{-1}\) to this equation. Similarly to the proof of (11.6) we compute
This leads us to consider the problem
where \({{\tilde{H}}}\) is a radial function satisfying
and
with
by (11.13).
We claim that there is a choice of \({{\tilde{H}}}\), which defines a linear operator of H, and satisfies
Indeed, the equation \(L_0[{{\tilde{H}}}] = H\) for radial functions has the form
We select the solution
Instead of (11.19) we consider
We then have the following estimate for \({\tilde{\Phi }}_1\):
For the proof of this we construct a barrier. First we find a solution to
The equation may be integrated explicitly, noting that
and that the constants are in the kernel of this operator. We then have
and this implies
Let
where \(\chi _0\in C^\infty ({\mathbb {R}})\), \(\chi _0(s)=1\) for \(s\le 1\) and \(\chi _0(s)=0\) for \(s\ge 2\). Define \({\tilde{\phi }}_1 = \frac{1}{\tau ^\nu (\log \tau )^m} \phi _1\chi \). We have
for some \(C_1>0\), \(\delta >0\) (assuming \(\tau _0\) large). Now consider
A computation, using (11.20), shows that
satisfies
for some \(c>0\). This step needs \(\nu -1<\frac{\varepsilon }{2}\) and \(\nu +\frac{\sigma }{2}<3\). By comparison, we find that \({\tilde{\Phi }}_1\) satisfies (11.23).
The solution \({\tilde{\Phi }}_1\) of (11.22) satisfies
Applying \(L_0\) to this equation we find that
satisfies
with
Let us verify that \(\Phi _1\) and \(H_1\) satisfy the conditions stated in (11.15), (11.16), (11.18). Indeed, from standard parabolic estimates and (11.23) we have
Differentiating in \(y_j\), \(j=1,2\) the equation (11.22) and using standard parabolic estimates, together with (11.21) and (11.25), we obtain
The definition \(\Phi _1 =L_0[{\tilde{\Phi }}_1]\) and the estimates (11.23), (11.25), (11.26) give the estimate (11.15).
We compute
Note that \(\int _{{\mathbb {R}}^2} \Phi _1(\cdot ,\tau )=0\). So, by a direct radial computation of \(\Psi _1 = (-\Delta )^{-1} \Phi _1\) and (11.15) we obtain
This estimate and the ones already obtained for \({\tilde{\Phi }}_1\) (11.25), (11.26) and for \(\Phi _1\) (11.15) yields
which is the desired estimate (11.16).
Finally, the zero mass condition (11.18) follows from the form of \(H_1\) (11.24) and its decay. The mass condition for \(\Phi _1\) (11.17) follows from \(\Phi _1 =L_0[{\tilde{\Phi }}_1]\) and the decay of \({\tilde{\Phi }}_1\) (11.23) and (11.25). \(\square \)
Next we would like to obtain a result similar to Proposition 10.1 for the problem (11.12). In order to simplify this step, we will modify this equation by allowing a parameter in the initial condition. This technical obstruction will be removed in the proof of Proposition 11.1. Thus we consider
where \({{\tilde{Z}}}_0\) is defined in (6.4).
The next result allows us to say that if in equation (11.27) the right hand side has fast decay, then we can decompose the solution similarly as in Proposition 10.1. This result is an extension of that proposition to an equation that has the extra operator A in it, which is treated as a perturbation.
Lemma 11.3
Let \(0<\sigma <1\), \(\varepsilon >0\), \(\sigma +\varepsilon <2\), \(1<\nu <\min (1+\frac{\varepsilon }{2},3-\frac{\sigma }{2},\frac{3}{2})\). Let \(0<q<1\). Then there is \(C>0\) such that for \(\tau _0\) sufficiently large and for H radially symmetric with \(\Vert H\Vert _{\nu ,m,4+\sigma ,\varepsilon }<\infty \) and
the solution \(\Phi \) to (11.27) can be decomposed as \(\Phi = \Phi _0 + \frac{a(\tau )}{2} Z_0\) with the estimates
Moreover \(\Phi _0\) and a are linear operators of H.
Proof of Lemma 11.3
We will treat the operator A as a perturbation and therefore consider
Let \(\Phi _1\), \(H_1\) be the functions constructed in Lemma 11.2. Setting \(\Phi = \Phi _1 + \Phi _2\), (11.28) is equivalent to the following equation for \(\Phi _2\)
We now apply Proposition 10.1 to (11.29). We have that \(\Vert H_1\Vert _{\nu ,m,6+\sigma ,\varepsilon }<\infty \) by (11.16), \(H_1\) is radial and satisfies the zero mass condition (11.18). By Proposition 10.1 and Lemma 10.10 there exists \(c_1\) such that the solution \(\Phi _2 \) of (11.29) satisfies
with the estimates
(We are ignoring the factor \(\frac{1}{(\log \tau _0)^{1-q}}\) in the estimate of \(a(\tau )\).) We also know that \(c_1\) is a linear function of \(H_1\) and satisfies
Combining (11.15) and (11.30) we conclude that \(\Phi \), the solution to (11.28), can be decomposed as
where \(\Phi _0(y,\tau ) = \Phi _1 + \Phi _2^\perp \) is radial and satisfies
and \(a(\tau )\) satisfies, combining (11.16) and (11.31),
We summarize the previous finding as follows. Given H radial satisfying \(\int _{{\mathbb {R}}^2} H(\cdot ,\tau )=0\) for \(\tau >\tau _0\) and \(\Vert H\Vert _{\nu ,m,4+\sigma ,\varepsilon } <\infty \), let us denote \(T_0(H) =\Phi _0 = \Phi _1 + \Phi _2^\perp \) and \(T_a(H) = a(\tau ) \) so that the solution \(\Phi \) of (11.28), is \(\Phi = \Phi _0 + \frac{a(\tau )}{2} Z_0 = T_0[H] + \frac{1}{2} T_a[H] Z_0\). Then \(T_0\), \(T_a\) are linear and have the estimates
where
Moreover \(c_1\) is a linear function of H and satisfies
We will apply these estimates to treat problem (11.27), which can be written as the fixed point problem
We claim that
for some \(\vartheta >0\), where C is independent of \(\tau _0\), and
Assume for the moment that (11.34), (11.35) hold. The we see that
For \(\tau _0\) large this gives
which is the desired result.
For the proof of estimates (11.34), (11.35) we will need the following property. If \(\Phi \) satisfies \(|\Phi (y)|\le \frac{1}{(1+|y|)^{2+\kappa }}\) for some \(\kappa >0\) and \(\int _{{\mathbb {R}}^2}\Phi dy=0\), then
Indeed,
and
To prove (11.34), let us write \(\Psi _0 = (-\Delta )^{-1} \Phi _0\). Then
Using the definition of \(L^{-1}\) given in Lemma 11.1 we have that
where
and \(\psi \) is the decaying solution to the Liouville equation
From the definition \(\Psi _0 = (-\Delta )^{-1} \Phi _0\) and using that \(\int _{{\mathbb {R}}^2} \Phi _0 dy=0\) we have
which gives the estimate
Then formula (11.37) gives
We note that by (11.36) we have \(\int _{{\mathbb {R}}^2} U g z_0 dy = 0\). Then, \(\psi \) has the estimate
It follows that \(A[ \Phi _0] = U g + U \psi \) satisfies
From this inequality we obtain (11.34).
The proof of (11.35) is similar. This time \(A[a Z_0] = U g_1 + U \psi _1\) where
and \(\psi _1\) is the radial decaying solution to
We then obtain that
From this estimate we deduce (11.35). \(\square \)
Before proving Proposition 11.1 as stated, we obtain a version of it for the problem
where
Lemma 11.4
Let \(0<\sigma <1\), \(\varepsilon >0\), \(\sigma +\varepsilon <2\) and \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{3}{2})\). Let \(0<q<1\). Then there is C such that for \(\tau _0\) large the following holds. Suppose that h is radially symmetric, satisfies \(\Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon }<\infty \) and
Then there exist \(c_1\in {\mathbb {R}}\) and a solution \(\phi (y,\tau )\) of problem (11.38) that define linear operators of h and satisfy
Proof
Consider equation (11.27), where H is the function constructed in Lemma 11.1. By Lemma 11.3, there is \(c_1\) such that the solution \(\Phi \) of (11.27) can be decomposed as \(\Phi = \Phi _0 + \frac{a(\tau )}{2} Z_0\), where \(\Phi _0\) and a satisfy the estimates stated in that proposition. In combination with (11.4) we find
Moreover \(\Phi _0\), a, \(c_1\) are linear operators of H.
From standard parabolic estimates and (11.39) we obtain
We consider the equation for \(\Phi _0 = \Phi - \frac{a(\tau )}{2}Z_0\), obtained from (11.27), and differentiate with respect to \(y_j\), \(j=1,2\). Using standard parabolic estimates, together with (11.39), (11.41), and the bound for \(a'(\tau )\) in (10.92), we obtain
Let us define \(\phi = L[\Phi ]\). Then \(\phi \) satisfies (11.38) because \(L[Z_0]=0\) and thanks to (11.39), (11.41), (11.42) we find
In the rest of the proof we show that
For this we consider the equation (11.38) written in the form
where
Using (11.43) and the radial formula for \(\psi = (-\Delta )^{-1} \phi \), we get
This estimate and the definition of the norm \( \Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon }\), give
We now construct a barrier very similar to the proof of Proposition 8.1
where \({{\tilde{g}}}_2\) is the function (10.60). We consider (11.44) in \(\{ \, (y,\tau ) \ | \ \tau> \tau _0, \ |y|> R_0\, \}\) where \(R_0>0\) is a large constant. For suitable constants \(A_1\), \(A_2\), \(A_3\), C the function \(C \Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon } {\bar{\phi }} \) is a supersolution. This computation requires \(\nu <\frac{3}{2}\).
Moreover \(\phi (y,\tau ) \le C \Vert h\Vert _{\nu ,m,6+\sigma ,\varepsilon } {\bar{\phi }}(y,\tau )\) at \(|y| = R_0\). The initial conditions also compare well. Indeed, by Lemma 10.11 and (11.40)
and this is supported on \(\rho \le 2 \sqrt{\tau }_0\), so
By the maximum principle
This finishes the proof. \(\square \)
Proof of Proposition 11.1
Let \({\hat{\phi }}\), \(c_1\) be the solution to (11.38) constructed in Lemma 11.4. Let \(\phi _1\) be the solution to (10.100). By Lemma 10.12\(\phi _1\) satisfies
where \(1<\nu _0<\frac{7}{4}\). Then the solution \(\phi \) to (8.9) that we construct is given by
To get the desired estimate on \(\phi \) we need to estimate \(|c_1 \phi _1|\). Let f be given by (10.12). By (11.40) and (11.45)
provided \(\frac{1}{2}+\nu -\nu _0<0\). However \(\nu _0\) can be taken close to \(\frac{7}{4}\), so we obtain the result by assuming \(\nu < \frac{5}{4}\) in addition to the other constraints needed in Lemma 11.4, namely \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{3}{2})\). \(\square \)
12 Linear Estimate with Second Moment (General)
A convenient property of problem (8.3) is that it can be split into Fourier modes. If we decompose
then \(\phi \) solves (8.3) if and only if \(\phi _i\) solves (8.3) where h is replaced with \(h_i\), for \(i=0,1\). If \(h=h_1\) we say that h has no radial mode.
For the proof Proposition 8.2 in the general case we will consider in a first step the equation (8.3) but without the operator B, namely,
for functions with no radial mode, as explained at the beginning of Sect. 11. Later on, we will consider equation (8.3) for functions with no radial mode, where we will treat the operator \(B[\phi ]\) as a perturbation term that can be assimilated to the right hand side.
The main step in the proof is the following estimate, valid when the functions involved have no radial mode:
Proposition 12.1
Let \(0<\sigma <1\), \(0<\varepsilon <2\), \(0<\nu <\min ( 1 + \frac{\varepsilon }{2}, \frac{3}{2}-\frac{\sigma }{2})\), \(m\in {\mathbb {R}}\). Then there is a \(C>0\) such that for any \(\tau _0\) sufficiently large the following holds. Suppose that \(h(y,\tau )\) has no radial mode and satisfies \(\Vert h\Vert _{\nu ,m,5+\sigma ,\varepsilon }<\infty \),
Then the solution \(\phi (y,\tau )\) of (12.3) satisfies
Proof
Since \(h(y,\tau )\) has no radial mode, all functions involved in the proof have also this property. We use the notation from §9.2, particular \(g = \frac{\phi }{U}-(-\Delta )^{-1}\phi \), \(g^\perp = g - a\) with \(a(\tau )\in {\mathbb {R}}\) such that
But
because g has no radial mode, so that \(a(\tau ) = 0\), \(g^\perp = g\), \(\phi ^\perp = \phi \). Then the proof proceeds as the proof of Proposition 10.1 with some simplifications, since there is no need to estimate a.
We write (12.3) as
We multiply this equation by g and integrate in \({\mathbb {R}}^2\).
Let \(R>0\) be a large fixed constant and let
Let \(T_2>\tau _0\) and let
The following estimates are valid for \(\tau \in [\tau _0,T_2]\). As in the proof of Proposition 10.1 we get
where
Similarly as in Lemma 10.8, from (12.6) we get
The proof is presented below. We use this to estimate
which implies
We deduce that
by choosing R as a large constant.
Now we let \(T_2\rightarrow \infty \) and find
The inequalities that follow hold for \(\tau >\tau _0\).
Combining (12.8) with (12.6) we obtain
and using (12.7) we also get
Let \(\psi = (-\Delta )^{-1}\phi \) so that \( \phi = U g + U \psi \). Using Lemma 9.1 and the previous estimate we obtain
We consider the equation (12.3) in \({\mathbb {R}}^2 {\setminus } B_{R}(0)\) written in the form
where
By (12.9) and the definition of the norm \( \Vert h\Vert _{\nu ,m,5+\sigma ,\varepsilon }\),
Here we are using \(\varepsilon <2\). Using barriers as in the proof of Lemma 10.8 we get
(For this we need \( \nu < 1+\frac{\varepsilon }{2}\), \(\nu + \frac{\sigma }{2} < \frac{3}{2}\).) This proves (12.5). \(\square \)
Proof of (12.7)
We define
which satisfies the equation
where
and
As in the proof of Lemma 10.7 we obtain
where
Applying parabolic estimates to (12.10) and a scaling argument we find
Using (12.11), (12.12) and \(g_0 = g U\) we get that
We observe that for \(i=1,2\)
Indeed,
But from \(g = \frac{\phi }{U} - \psi \), \( \psi = (-\Delta )^{-1}\phi \) we have
Multiplying this equation by \(z_i = \nabla \Gamma _0 e_i\) defined in (9.2) and integrating we get
which is the desired claim (12.13). We note that
Now we can apply Remark 9.1 and deduce that for any \(\vartheta \in (0,1)\) there is C such that
We next estimate \({{\tilde{h}}}\). From Remark 9.1 and the assumptions on h, in particular (12.4), we have
for any \(\vartheta \in (0,1)\). Also from (12.11) we have
Therefore, from (12.15), (12.11), (12.14) we find that, for any \(\vartheta >0\),
We now use a barrier as in the proof of Lemma 10.8, in a domain of the form \(({\mathbb {R}}^2 {\setminus } B_{R_0} ) \times (\tau _0,\infty )\) where \(R_0\) is a large constant. We let \({{\tilde{g}}}(y)\) be the radial decaying solution to \(-\Delta _6 {{\tilde{g}}} = \frac{1}{(1+|y|)^{5+\sigma }}\) and
where
We assume that \(\nu <\frac{3}{2}-\frac{\sigma }{2}-\frac{\vartheta }{2}\), \( \nu < 1+\frac{\varepsilon }{2}\), \(\nu + \frac{\sigma }{2} < \frac{3}{2}\), and \(\sigma +\vartheta <1\). Since \(\vartheta >0\) is arbitrary we only need \(\nu <\frac{3}{2}-\frac{\sigma }{2}\), \( \nu < 1+\frac{\varepsilon }{2}\) and \(\sigma <1\). Then, for an appropriate choice of \(C_1\), \(C_2\), the function \(R K {{\bar{g}}}(y,\tau )\) is a supersolution. By the maximum principle
This proves the desired estimate (12.7). \(\square \)
Next we consider equation (8.3), which we recall,
For \( \phi \) with no radial mode we can write
Corollary 12.1
Let \(0<\sigma <1\), \(0<\varepsilon <2\), \(1<\nu <\min ( 1 + \frac{\varepsilon }{2}, \frac{3}{2}-\frac{\sigma }{2})\), \(m\in {\mathbb {R}}\). Then there is a \(C>0\) such that for any \(\tau _0\) sufficiently large the following holds. Suppose that \(h(y,\tau )\) has no radial mode and satisfies \(\Vert h\Vert _{\nu ,m,5+\sigma ,\varepsilon }<\infty \). Then there is a solution \(\phi (y,\tau )\), \(\mu _j\) of (12.16) that is a linear operator of h and satisfies
Proof
Using Proposition 12.1, there is a linear operator T so that given h with \( \Vert h\Vert _{\nu ,m,5+\sigma ,\varepsilon }<\infty \), with no radial mode, and satisfying the condition (12.4) associates the solution \(\phi \) of (12.3). Then the solution \(\phi \) of (12.16) can be written as
where \(\mu _j\) is chosen so that
The estimate (12.5) implies
Using standard parabolic estimates we also get
To estimate \(\mu _j\) note that multiplying (12.16) by \(y_j\) and integrating we get that
Therefore
and from the definition (12.19)
We also have that
Then for \(\tau _0\) large we deduce the estimate (12.17).
Finally, from (12.19) we get (12.18) with \({\tilde{\mu }}_j\) a linear operator of h satisfying
\(\square \)
We are now in a position to prove Proposition 8.2 in the general case.
Proof of Proposition 8.2
We decompose \(h = h_0 + h_1\) and \(\phi = \phi _0 + \phi _1\) as in (12.1), (12.2). We apply Proposition 11.1 to get
To estimate \(\phi _1\) we use Corollary 12.1. First we select \(0<\vartheta <\frac{1}{2}\). Then note that
Then Corollary 12.1 gives a solution \(\phi _1\) of (12.16) such that
We take \({\bar{\sigma }}=1-\vartheta \) and \({\bar{\varepsilon }}=\varepsilon +\sigma +\vartheta \) and get
and
because \(h_0\) is radial, with
But
and hence
To apply Corollary 12.1 we need \(1<\nu <1+\frac{\varepsilon }{2}\) and \(\nu <1+\frac{\vartheta }{2}\). Given \(1<\nu < \min ( 1+\frac{\varepsilon }{2},3-\frac{\sigma }{2}, \frac{5}{4})\) we can select \(\vartheta \in (0,\frac{1}{2})\) such that \(\nu <1+\frac{\vartheta }{2}\) and then proceed. This concludes the proof. \(\square \)
13 The Outer Problem
We consider the linear outer problem
where
For \(g:{\mathbb {R}}^2 \times (t_0,\infty )\rightarrow {\mathbb {R}}\) we consider the norm \(\Vert g \Vert _{**,o}\) defined as the least K such that for all \((x,t)\in {\mathbb {R}}^2\times (t_0,\infty )\)
where \(A>0\) is a constant.
We also define the norm \(\Vert \phi \Vert _{*,o}\) as the least K such that
for all \((x,t)\in {\mathbb {R}}^2\times (t_0,\infty )\).
We assume that the parameters a, b satisfy the constraints
There is no restriction on \(\beta \).
We recall from (4.1) that we are assuming that
and
where \(0<\sigma <\frac{1}{2}\).
Proposition 13.1
Assume that a, b satisfy (13.2), \(\frac{A}{\lambda (t_0)^2}\) is sufficiently large, and \(\lambda , \xi \) satisfy (13.3), (13.4). Then there is a constant C so that for \(t_0\) sufficiently large and for \(\Vert g\Vert _{**,o}<\infty \) there exists a solution \(\phi ^o= {\mathcal {T}} ^o_{{{\textbf {p}}}}[g]\) of (13.1), which defines a linear operator of g and satisfies
Proposition 6.3 in Sect. 6 follows from Proposition 13.1 with \(A = t_0\).
Lemma 13.1
Let \(2<\beta <6\) and h(r) satisfy
where \(\lambda >0\). Then there is a unique bounded radial function \(\varphi (r)\) satisfying
Moreover \(\varphi \) satisfies
Proof
The equation for \(\varphi \) is given by
We change variables \(\rho = \frac{r}{\lambda }\) and let \(\varphi (r) = {\bar{\varphi }}(\frac{r}{\lambda })\). Then we need to solve
where
By (13.5)
The bounded solution is given by
By direct computation we get
and this implies (13.6). \(\square \)
Proof of Proposition 13.1
To find a pointwise estimate for the solution \(\phi ^o\) we construct a barrier.
Using polar coordinates \(x-\xi (t) = r e^{i\theta }\), \(L^o\) can be written as:
First we construct a function \({\tilde{\psi }}(r,t)\) such that
Let
Choosing a large constant \(C_1\), \(\psi _1\) satisfies
where \(c>0\). Here we require \(a<4\) and \( a < 1+ \frac{b}{2}\), which are part of the conditions (13.2). Then
but
and so
We note that for \(r\le \sqrt{t-t_0+A}\) we have
where we have used that \(A\ge \lambda (t)^2\).
Let \({\tilde{\psi }}_2(r;\lambda )\) be the bounded solution of
given by Lemma 13.1. Then \({\tilde{\psi }}_2\) can be written as
for a function \({\bar{\psi }}_2\) satisfying
Let
Then, using (13.10) and (13.3), we get
Therefore there is \(\delta >0\) (fixed independent of \(t_0\)) such that for all \(t_0\) large,
Let \(\chi _0 \in C^\infty ({\mathbb {R}})\) be such that \(\chi _0(s) = 1 \) if \(s\le 1\) and \(\chi _0(s)=0\) if \(s\ge 2 \) and define
We consider
where \(M>0\) is a constant to be fixed later. We compute, using (13.7)
where
We have, by (13.10),
where \(C_2\) is independent of M (although it depends on \(\delta \)), and is supported on \(\delta \sqrt{t-t_0+A} \le r \le 2 \delta \sqrt{t-t_0+A}\).
We claim that there is \(M>0\) and \({{\tilde{c}}}>0\) so that for all \(t_0\) sufficiently large
for all \(r>0\), \(t>t_0\).
Indeed, if \(r\le \delta \sqrt{t-t_0+A}\), then from (13.12), (13.7), (13.11) and (13.9) we get
if \(M\ge C\). Here we fix \(M=C\).
If \( \delta \sqrt{t-t_0+A} \le r\le 2\delta \sqrt{t-t_0+A}\), then by (13.12), (13.7), (13.9) and (13.13) we get
By taking \(\frac{A}{\lambda (t_0)^2}\) large, we get
for \( \delta \sqrt{t-t_0+A} \le r\le 2\delta \sqrt{t-t_0+A}\).
If \(r\ge 2\delta \sqrt{t-t_0+A}\), by (13.12) and (13.8)
if \(\frac{A}{\lambda (t_0)^2}\) is sufficiently large.
Combining (13.15), (13.16) and (13.17) we deduce the estimate (13.14).
Let
Then, by (13.14),
but
Using (13.4) we see that if \(t_0\) is sufficiently large,
\(\square \)
A direct consequence of the proof of Proposition 13.1 (using the same barriers) is the following, for the initial value problem
Consider the norm
where \(b \in (2,6)\), \(A>0\).
Proposition 13.2
Assume that a, b satisfy (13.2), \(\frac{A}{\lambda (t_0)^2}\) is sufficiently large, and \(\lambda , \xi \) satisfy (13.3), (13.4). Then there is a constant C so that for \(t_0\) sufficiently large and for \(\Vert \phi ^o_0 \Vert _{*,b}<\infty \) there exists a solution \(\phi ^o\) of (13.18), which defines a linear operator of \( \phi ^o_0\) and satisfies
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Acknowledgements
J. Dávila has been supported by a Royal Society Wolfson Fellowship, UK. M. del Pino has been supported by the Royal Society Research Professorship Grant RP-R1-180114 and by the ERC/UKRI Horizon Europe Grant ASYMEVOL, EP/Z000394/1. J. Dolbeault thanks the ANR Project Conviviality # ANR-23-CE40-0003 for partial support. M. Musso has been supported by EPSRC research Grant EP/T008458/1. We are grateful to Federico Buseghin for his valuable comments and corrections.
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Dávila, J., del Pino, M., Dolbeault, J. et al. Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System. Arch Rational Mech Anal 248, 61 (2024). https://doi.org/10.1007/s00205-024-02006-7
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DOI: https://doi.org/10.1007/s00205-024-02006-7