Abstract
We consider the semilinear heat equation
with \(f(u)=|u|^{p-1}u\log ^a (2+u^2)\), where \(p>1\) is Sobolev subcritical and \(a\in {\mathbb {R}}\). We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely \(u' =|u|^{p-1}u\log ^a (2+u^2)\). In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.
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1 Introduction
1.1 Motivation of the Problem
This paper is devoted to the study of blow-up solutions for the following semilinear heat equation:
Here \(u(t):x\in {{\mathbb {R}}^N} \rightarrow u(x,t)\in {{\mathbb {R}}}\) with focusing nonlinearity f defined by:
We assume in addition that \(p>1\) and if \(N\geqq 3\), and we further assume that
Note that when \(a\ne 0\), the nonlinear term is not homogeneous, and this is the focus of our paper.
By standard results the problem (1.1) has a unique solution for any \(u_0\in L^{\infty }({\mathbb {R}}^N)\). More precisely, there is a unique maximal solution on [0, T), with \(T\leqq \infty \). If \(T<\infty \), we say that the solution of (1.1) blows up in finite time. In that case, it holds that \(\Vert u(t)\Vert _{L^{\infty }({\mathbb {R}}^N)}\rightarrow \infty \) as \(t\rightarrow T\). Such a solution u is called a blow-up solution of (1.1) with the blow-up time T.
In the case \(a=0\), equation (1.1) reduces to the semilinear heat equation with power nonlinearity:
In the literature, the determination of the blow-up rate has been linked to the terminology of “Type I/Type II solutions”, first introduced (up to our knowledge) by Matano and Merle in [19]. In that paper, if a solution u to (1.4) blows up at time T and satisfies for all \(t\in [0,T)\),
for some positive constant C, independent of time t, then u is called a Type I. If not, then u is said to be of Type II. Note that the bound given in (1.5) is (up to a multiplying factor) a solution of the associated ODE \(u'=u^p\).
In the subcritical case under consideration (1.3), we know from Giga and Kohn [6,7,8], and also Giga, Matsui and Sasayama [9] that all blow-up solutions of (1.4) are of Type I. Moreover, from the construction provided by Nguyen and Zaag [22], we know that Type I solutions are available for any superlinear exponent p, not only in the subcritical case, despite what the authors noted at that time.
As for Type II solutions, we know that they are available in the critical range (see Schweyer [25], Harada [18], Del Pino, Musso and Wei [3], Collot, Merle and Raphaël [2], Filippas, Herrero and Velàzquez [5]), and also in the supercritical range (see Herrero and Velàzquez [17], Mizoguchi [20], Seki [26, 27].
Going back to the proof given in [9] for the fact that all blow-up solutions for equation (1.4) in the subcritical range (1.3) are of Type I, we would like to mention that the following estimate is central in the argument:
there w is the similarity variables version of the solution defined in (1.17) below and \({\mathbf {B}}_R\equiv B(0,R)\) is the open ball of radius R centered at the origin in \({\mathbb {R}}^N.\)
Exploiting the non-trivial perturbative method introduced by the authors in [13, 14] in the hyperbolic case and arguing as in the non perturbed case in [9], Nguyen proved in [21] a similar result to (1.5), valid in the subcritical case, for a class of strongly perturbed semilinear heat equations
under the assumptions \(|h(u)|\leqq M(1+|u|^p)\log ^{-a}(2+u^2)\), for some \(M > 0\) and \(a>1\). Obtaining the same blow-up rate is reasonable, since the dynamics is still governed by the ODE \(u' =|u|^{p-1}u\). Furthermore, the proof remains (non trivially) perturbative with respect to the homogeneous PDE (1.4), which is scale invariant.
This leaves unanswered an interesting question: is the scale invariance property crucial in deriving the blow-up rate?
In fact we had the impression that the answer was ”yes”, since the scaling invariance induces in similarity variables a PDE which is autonomous in the unperturbed case (1.4), and asymptotically autonomous in the perturbed case (1.7).
In this paper we prove that the answer is ”no” from the example of the non homogeneous PDE (1.4). In fact, our situation is different from (1.4) and (1.7). Indeed, the term \(|u|^{p-1}u\log ^a (2+u^2)\) is playing a fundamental role in the dynamics of the blow-up solution of (1.1). More precisely, we obtain an analogous result to (1.5) but with a logarithmic correction as shown in (1.28) below. In fact, the bow-up rate is given by the solution of the associated ODE \(u'=|u|^{p-1}u\log ^a (2+u^2)\).
In this paper, we study the blow-up rate of any singular solution of (1.1). Before handling the PDE, we first consider the ODE associated to (1.1),
and show that the nonlinear term including the logarithmic factor gives rise to different dynamics. In fact, thanks to [4] (see Lemma A1), we can see that the solution \(v_T\) satisfies
and
Therefore, it is natural to extend the terminology “Type I/Type II solutions” for the blow-up of a solution u(x, t) of (1.1) by the following:
Let us mention that Duong, Nguyen and Zaag construct in [4] a solution of equation (1.1) which blows up in finite time T, only at one blow-up point \(x_0\), according to the following asymptotic dynamics:
Here \(v_T(t)\) is the solution of (1.8) with an equivalent given in (1.9). Note from (1.13) that the constructed solution is of Type I.
Concerning the blow-up rate for the hyperbolic equations with a non-homogeneous main term, we would like to mention that in [15] and [16], we consider the semilinear wave equation
where \(a\in {\mathbb {R}}\) and \(p>1\) is subconformal, in the sense that \((N-1)p<N+3\). We prove that the exact blow-up rate of any singular solution of (1.14) is given by the ODE solution associated with (1.14), namely
Let us mention that the nonlinear term involving the logarithmic factor gives raise to different dynamics. To be precise, the solution \(V_T\) satisfies
Since the blow-up rate is given by \(V_T(t)\), we see that the effect of the nonlinearity is completely encapsulated in (1.16). Note that before [15, 16], we could successfully implement our perturbative method in [10,11,12,13,14] to derive the blow-up rate for some classes of perturbed wave equations where the main nonlinear term is power-like (hence, homogeneous).
1.2 Strategy of the Proof
Going back to the equation under study in this paper (see (1.1) and (1.2)), we introduce the following similarity variables, defined for all \(x_0\in {\mathbb {R}}^N\):
Here \(\psi _T(t)\) is the explicit rate given in (1.10). On may think that it would be more natural to replace \(\psi _T(t)\) by \(v_T(t)\) (defined in (1.8)) in this definition, since the latter is an exact solution of the ODE (1.8). That might be good, however, as \(v_T(t)\) has no explicit expression, the calculations will immediately become too complicated. For that reason, we preferred to replace the non-explicit \(v_T(t)\) by its explicit equivalent \(\psi _T(t)\) in (1.10). The fact the latter is not an exact solution of (1.8) will have no incidence in our analysis.
From (1.1) and (1.17), the function \(w_{x_0,T}\) (we write w for simplicity) satisfies, for all \(y\in {\mathbb {R}}^N\) and \(s\geqq \max ( -\log T,1)\),
where
and
In the new set of variables (y, s), studying the behavior of u as \(t \rightarrow T\) is equivalent to studying the behavior of w as \(s \rightarrow +\infty \).
While reading Giga and Kohn [6,7,8] dedicated to the blow-up rate of the homogeneous case (1.4), one sees that the existence of a Lyapunov functional for the similarity variables’ version (1.18) with \(a=0\) is central in the argument. Clearly, the invariance of equation (1.4) under the scaling transformation \(u \mapsto u_{\lambda }(x,t)=\lambda ^{\frac{1}{p-1}}u(\lambda x,\lambda ^2 t)\) was crucial in the construction of the Lyapunov functional. The fact that equation (1.1) is not invariant under the last scaling transformation implies that the existence of a Lyapunov functional in similarity variables is far from being trivial (see [21, 23] in the parabolic case and [10,11,12,13,14,15] in the hyperbolic case).
In this paper, we construct a Lyapunov functional in similarity variables for the problem (1.18). Then, we prove that the blow-up rate of any singular solution of (1.1) is given by the solution of (1.8).
Let us explain how we derive the Lyapunov functional. As we did for the perturbed wave equation with a conformal exponent in [10, 12, 13], we proceed in 2 steps:
-
Step 1: we first introduce some functional (not a Lyapunov functional) for equation (1.18), which is bounded by \(s^\alpha \) for some \(\alpha >0\), then show that w enjoys also a polynomial (in s) bound.
-
Step 2: then, viewing equation (1.18) as a perturbation of the case of a pure power nonlinearity (case where \(a=0\) in (1.18)) by the following terms:
$$\begin{aligned} \frac{a}{(p-1)s} w \quad \text { and} \quad e^{-\frac{ps}{p-1}}s^{\frac{a}{p-1}} f(\phi (s)w), \end{aligned}$$(1.21)
we use the rough estimates on w proved in the first step, in order to control the \(\ll \) perturbative \(\gg \) terms in (1.18). This way, we find a Lyapunov functional for (1.18), then use it to prove that the solution itself is bounded.
Specifically, in Step 1, we would like to add the following regarding the effect of the perturbation terms (1.21) and the way we handle them: The first term is a lower order term which was already handled in the Sobolev subcritical perturbative case treated in [21, 23]. However, since the nonlinear term \(e^{-\frac{ps}{p-1}}s^{\frac{a}{p-1}} f(\phi (s)w)\) depends on time s, we expect the time derivatives to be delicate. Thanks to the fact that \(uf(u)-(p+1)\int _0^uf(v) {\mathrm {d}}v\sim \frac{2a}{p+1}|u|^{p+1}\log ^{a-1}(2+u^2)\), as \(u\rightarrow \infty \), we construct a functional (in Section 2) satisfying the following kind of differential inequality:
this implies a polynomial estimate.
In order to state our main result, we start by introducing the functionals
where
Moreover, for all \(s\geqq \max (- \log T,1 )\), we define the functional
where \(\theta \) is a sufficiently large constant that will be determined later. We derive that the functional L(w(s), s) is a decreasing functional of time for equation (1.18), provided that s is large enough. Clearly, by (1.23), (1.24) and (1.26), the functional L(w(s), s) is a small perturbation of the natural energy E(w(s), s).
Our main theorem in this paper is as follows:
Theorem 1
(A Lyapunov functional in similarity variables) Consider u a solution of (1.1), with blow-up time \(T>0\). Then, there exists \(t_1\in [0,T) \) such that,
for all \(s\geqq -\log (T-t_1)\) and \(x_0\in {\mathbb {R}}^N\), we have
where \(w=w_{x_0,T}\) is defined in (1.17).
Remark 1.1
We choose to put forward this result proving the existence of a Lyapunov functional and state it as the first result of our paper (namely Theorem 1), mainly because we consider it as the crucial step in our argument, and also because its proof is far from being trivial.
The existence of this Lyapunov functional L(w(s), s) together with a blow-up criterion for equation (1.18) make a crucial step in the derivation of the blow-up rate for equation (1.1). Indeed, with the functional L(w(s), s), we are able to adapt the analysis performed in [6,7,8] for equation (1.4) and obtain the following result:
Theorem 2
(Blow-up rate for equation (1.1)) Consider u a solution of (1.1), with blow-up time \(T>0\). Then, there exists \(t_2\in [t_1,T) \) such that for all \(t\in [t_2,T)\), we have
where \(K=K(p, a, T, t_2,\Vert u(\tilde{t_2})\Vert _{ L^{\infty }})\), for some \(\tilde{t_2}\in [0,t_2)\).
Remark 1.2
Note that the blow-up rate in this upper bound is sharp, since we have, from a simple comparison argument, the lower bound
where the last equivalence was given in (1.9).
Remark 1.3
Let us remark that we can obtain the same blow-up rate for the more general equation
under the assumption that \(|k(u)|\leqq M(1+|u|^{p}\log ^b(2+u^2))\), for some \(M > 0\) and \(b<a-1\). More precisely, under this hypothesis, we can construct a suitable Lyapunov functional for this equation. Then, we can prove a similar result to (1.28). However, the case where \(a-1\leqq b<a\) seems to be out of reach of our technics, though we think we may obtain the same rate as in the unperturbed case.
This paper is organized as follows: in Section 2, we obtain a rough control of the solution w. In Section 3, thanks to that result, we prove that the functional L(w(s), s) is a Lyapunov functional for equation (1.18). Thus, we get Theorem 1. Finally, by applying this last theorem, we give the proof of Theorem 2.
Throughout this paper, C denotes a generic positive constant depending only on p, N and a, which may vary from line to line. As for M, it will be used for constants depending on initial data, in addition to p, N and a. We may also use \(K_1\), \(K_2\), \(K_3\)... \(M_1\), \(M_2\), \(M_3\)... \(Q_1\), \(Q_2\), \(Q_3\) for constants having the same dependence as M. If necessary, we may write explicitly the dependence of the constants we use.
Moreover, we denote by \({\mathbf {B}}_R\) the open ball in \({\mathbb {R}}^N\) with center 0 and radius R. Finally, note that we use the notation \(f(s)\sim g(s)\) when \(\displaystyle {\lim _{s\rightarrow \infty }\frac{f(s)}{g(s)}=1}\).
2 A Polynomial Bound for Solutions of Equation (1.18)
This section is devoted to the derivation of a polynomial bound for a global solution of equation (1.18). More precisely, we have
Proposition 2.1
Let \(R>0\). Consider w a global solution of (1.18). Then, there exist \({\widehat{S}}_1={\widehat{S}}_1(a,p,N,R) \geqq 1\) and \(\mu =\mu (a,p,N,R)>0\) such that, for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), we have
where \(K_1\) depends on p, a, N, R and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}\).
Remark 2.1
By using the Sobolev’s embedding and the above proposition, we can deduce that for all \(r\in [2,2^*)\), where \(2^*=\frac{2N}{N-2}\), if \(N\geqq 3\) and \(2^*=\infty \), if \(N=2\),
where \(K_2(R)\) depends on p, a, N, R and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}\).
In order to prove this proposition, we need to construct a Lyapunov functional for equation (1.18). Accordingly, we start by recalling from (1.23) the functional
where F is given by (1.25). Then, we introduce the functionals
where \(m >0\) is a sufficiently large constant that will be fixed later.
In fact, the main target of this section is to prove, for some \({m}_0\) large enough, that the energy \(H_{{m}_0}(w(s),s)\) satisfies the inequality
which implies that \(H_{m _0} (w(s),s)\) satisfies the following polynomial estimate:
for some \(A_0>0\) and \(\mu _0>0\).
2.1 Classical Energy Estimates
In this subsection, we state two lemmas which are crucial for the construction of a Lyapunov functional. We begin with bounding the time derivative of E(w(s), s) in the following lemma:
Lemma 2.2
For all \(s \geqq \max (-\log T,1)\), we have
where \(\Sigma _{1}(s)\) satisfies
Proof
Consider \(s \geqq \max (-\log T,1)\). Multiplying (1.18) by \(\partial _{s} w\!\ \rho (y)\) and integrating over \({\mathbb {R}}^N,\) we obtain
Now, we control the terms \(\Sigma _{1}^{1}(s)\), \(\Sigma _{1}^{2}(s)\) and \( \Sigma _{1}^{3}(s)\). By using the following basic inequality
we write
Let us introduce the functions \(F_1\) and \(F_2\) defined by
and
By the expressions of \(F_1\), \(F_2\) given by (2.13) and (2.14) and the estimates (B.5) and (B.6), we obtain
which implies
From the expression of \(\phi = \phi (s)\) defined in (1.20), we have
Thus, using (2.16) and (2.17), we obtain
Similarly, by (B.4) and (2.17), we easily obtain
The results (2.8) and (2.9) follows immediately from (2.10), (2.12), (2.18) and (2.19), which ends the proof of Lemma 2.2.
Remark 2.2
By showing the estimate proved in Lemma 2.2, related to the so called natural functional E(w(s), s), we have some nonnegative terms in the right-hand side of (2.8) and this does not allow to construct a decreasing functional (unlike the case of a pure power nonlinearity). The main problem is related to the nonlinear term
To overcome this problem, we adapt the strategy used in [10,11,12,13,14, 21]. Indeed, by using the identity obtained by multiplying equation (1.1) by \(w\rho (y)\), then integrating over \({\mathbb {R}}^N\), we can introduce a new functional \( H_m(w(s),s)\) defined in (2.5), where \(m>0\) is sufficiently large and will be fixed such that \(H_{m}(w(s),s)\) satisfies a differential inequality similar to (1.22).
We will prove the following estimate on the functional J(w(s), s):
Lemma 2.3
For all \(s \geqq \max (-\log T,1)\), we have
where \(\Sigma _2(s)\) satisfies
Proof
Consider \(s \geqq \max (-\log T,1)\). Note that J(w(s), s) is a differentiable function and that we get
From equation (1.18) and the identity (2.17), we conclude that
According to the expressions of E(w(s), s), \(\phi (s)\) defined in (2.3) and (1.20) and the identity (2.17) with some straightforward computation, we obtain (2.20) where
and
Thanks to (2.17) and (2.15), we deduce
Hence, collecting (2.22) and (2.23), one easily obtains that \(\Sigma _{2}(s)\) satisfies (2.21), which ends the proof of Lemma 2.3.
2.2 Existence of a Decreasing Functional for Equation (1.18)
In this subsection, by using Lemmas 2.2 and 2.3 , we will construct a decreasing functional for equation (1.18). Let us define the functional
where \(H_{m}(w(s),s)\) is defined in (2.5), and m together with \(A=A(m)\) are constants that will be determined later.
We now state the following proposition:
Proposition 2.4
There exist \(m_0>1\), \(A (m_0) >0\), \(S_{1}\geqq 1\) and \(\lambda _1>0\), such that for all \(s=s_1 \geqq \max (-\log T,S_1)\), we have
where
Moreover, there exists \(S_{2}\geqq S_1\) such that for all \(s \geqq \max (-\log T,S_2)\), we have
Proof
From the definition of \(H_m(w(s),s)\) given in (2.5), Lemmas 2.2 and 2.3, we can write, for all \(s\geqq \max (-\log T,1)\),
where \(C_0\) stands for some universal constant depending only on N, p and a. We first choose \(m_0\) such that \(\frac{m_0(p-1)}{4(p+1)}-C_0=0\), so
We now choose \(S_1=S_1(a,p,N)\) large enough (\(S_1\geqq 1\)), so that for all \(s\geqq S_1\), we have
Then, we deduce that for all \(s\geqq \max (-\log T,S_1)\),
where \(\lambda _0=\inf ( \frac{m_0}{8}, \frac{m_0(p-1)}{4(p+1)} )\).
By using the definition of \(N_{m_0}(w(s),s)\) given in (2.24) together with the estimate (2.29), we easily prove that for all \(s\geqq \max (-\log T,S_1)\),
We now choose \(A(m_0) =C_0(m_0+1){S_1}^{-b}\), so we have
By integrating in time between s and \(s+1\) the inequality (2.30) and using (2.31), we easily obtain (2.25). This concludes the proof of the first part of Proposition 2.4.
We prove (2.27) here. Arguing by contradiction, we assume that there exists \(\tilde{s_1} \geqq \max (-\log T,S_2)\) such that \(N_{m_0}(w(\tilde{s_1}),\tilde{s_1})<-1\), where \(S_2\geqq S_1\) is large enough.
Now, we consider
where b is defined in (2.26). Thanks to (2.20) and (2.5), we have for all \(s \geqq \max (-\log T,1)\)
Let us choose \(S_2=S_2(a,p,N)\) is large enough, such that \(1-\frac{C_1}{S_2}\geqq \frac{1}{2}\). So, we write for all \(s \geqq \max (-\log T,S_2)\)
Since the energy \(N_{m_0}(w(s),s)\) decreases in time, we have \(N_{m_0}(w(s),s)<-1\), for all \(s\geqq \tilde{s_1}\). Then, for all \(s\geqq \tilde{s_1}\)
Thanks to (B.4), (B.10) and (2.17), we get for all \(s\geqq \tilde{ s_1}\)
Therefore, by using (2.34) and (2.35), there exist \({\tilde{S}}_2\geqq S_2\) large enough such that \( p+2-\frac{C}{({\tilde{S}}_2)^b}>0\), we have for all \(s\geqq \max (\tilde{s_1},{\tilde{S}}_2)\)
Thanks to Jensen’s inequality, we infer
This quantity must then tend to \(\infty \) in finite time, which is a contradiction. Thus (3.38) holds. This concludes the proof of Proposition 2.4.
2.3 Proof of Proposition 2.5.
Based on Proposition 2.4, a bootstrap argument given in [24], we are able to adapt the analysis performed in [9], to prove the following key proposition:
Proposition 2.5
For all \(q \geqq 2\), \(\varepsilon >0\) and \(R>0\) there exist \(\varepsilon _1=\varepsilon _1(q,R)>0\), \(\mu _1(q,R,\varepsilon )>0\) and \(S_{3}(q,R,\varepsilon ) \geqq S_2\) such that, for all \(s \geqq \max (-\log T, S_3),\) we have
where \(K_{3}(q,R,\varepsilon )\) depends on \( p, a, N,q, R,\varepsilon , s_1 =\max (-\log T,S_1)\) and \(\Vert w( {s}_1)\Vert _{H^1}\).
To prove Proposition 2.5, we will proceed as in [9]. In fact, by using Proposition 2.4, we easily obtain the following Corollary
Corollary 3
For all \(s\geqq \max (-\log T,S_2),\) we have
where b is defined in (2.26) and where \(K_4, K_5,K_6,... K_{12} \) depend on \( p, a, N, s_1 =\max (-\log T,S_1)\) and \(\Vert w( s_1)\Vert _{H^1}\).
Remark 2.3
Let us mention that, the estimates obtained in the above corollary are similar to the ones obtained in the pure power case treated in [9] except for the following features:
-
The presence of the term \(Ks^{b+1}\) instead of K.
-
In some estimates, we have the term F(u) instead of \(\frac{|u|^{p+1}}{p+1}\) in the pure power case. We easily overcome this problem thanks to the fact that \(uf(u)-(p+1)\int _0^uf(v) {\mathrm {d}}v\sim \frac{2a}{p+1}|u|^{p+1}\log ^{a-1}(2+u^2)\), as \(u\rightarrow \infty \).
In order to prove Proposition 2.5, we introduce the following local functional:
where \(\psi \in {\mathscr {C}}_0^\infty (\mathbb {R}^N)\)
satisfies
where \(R>0\). An argument similar to that in [9], implies the following estimate:
Proposition 2.6
There exist positive constants \( K_{13}=K_{13}(R)>0\) and \(S_{4}\geqq S_2\) such that, for all \(s \geqq \max (-\log T, S_4),\) we have
where \( K_{13}\) depends on \( p, a, N, R, s_1 =\max (-\log T,S_1)\) and \(\Vert w( s_1)\Vert _{H^1}\).
Proof
Most of the steps of the proof are the same as in the pure power case treated in [9] and some others are more delicate. For that reason, we leave the proof to Appendix C.
With Proposition 2.6, we are in a position to claim the following:
Lemma 2.7
There exists a positive constant \( K_{14}(R,\varepsilon )>0\) such that, for all \(s \geqq \max (-\log T, S_4)\)
where \( K_{14}(R,\varepsilon )\) depends on \( p, a, N, R, \varepsilon , s_1 =\max (-\log T,S_1)\) and \(\Vert w( s_1)\Vert _{H^1}\).
Proof
From (2.50) and the definition of \({\mathscr {E}}_\psi \) in (2.48), we have for all \( s \geqq \max (-\log T, S_4),\)
By exploiting (B.10), we write for all \(s \geqq \max (-\log T, S_4),\)
Thus, (2.51) follows from (2.53) and the property of \(\psi \). This conclude the proof of Lemma 2.7
By (2.51), the proof of estimate \((A_{q,R,\varepsilon })\) is available when we have
for some \(\mu _2(q,R,\varepsilon )>(b+1)q\). Note from (2.46) that (2.54) already holds in the case \(q = 2\).
In order to derive (2.54) for all \(q > 2\), we need the following result:
Lemma 2.8
There exist positive constants \( K_{16}(R)>0\) and \(S_5\geqq S_4\) such that, we have
Proof
Multiplying equation (1.18) with \(w\rho (y)\psi ^2,\) integrating over \(\mathbb {R}^N\) and using the definition of \({\mathscr {E}}_\psi (w(s),s)\) given in (2.48), we write
From (2.50), (C.12) and (2.42) we infer for all \(s \geqq \max (-\log T, S_4),\)
According to the the estimates (2.15) and the identity (2.17), we get for all \(s \geqq \max (-\log T, S_4),\)
Hence, using (2.56), (2.57) and (2.58), yields for all \(s \geqq \max (-\log T, S_4),\)
Taking \(S_5\geqq S_4\) large enough such that \(1-\frac{(p+1)C_2}{2S_5}>0\), we have for all \(s \geqq \max (-\log T, S_5),\)
Thus, (2.55) follows from the property of \(\psi \). This ends the proof of Lemma 2.8.
Now, we are ready to give the proof of Proposition 2.5.
Proof of Proposition 2.5:
[Proof of (2.54) for all \(q \geqq 2 \) by a bootstrap argument] The proof is obtained by following the same part in [9]. However, as explained before (see Remarks 2.3), in our case we have two additional problems. Let \(R>0\) and suppose that we have
for some \(\mu _2(q,4R,\varepsilon )>0\) and for some \(q \geqq 2\).
Combining (2.60) and (2.51), we write for all \(s \geqq \max (-\log T, \tilde{S_3})\),
for some \(\mu _3(q,R,\varepsilon )>\mu _2(q,R,\varepsilon )\). where \(\tilde{S_3}=\max (S_3,S_5)\). Thus, we use (2.40), (2.61) and apply Lemma A.1 with \(\alpha = q(p-\varepsilon +1)\), \(\beta =p-\varepsilon +1\), \(\gamma = \delta = 2\) to get that for all \(s \geqq \max (-\log T, \tilde{S_3})\),
for some \(\mu _4(q,R,\varepsilon )>\mu _3(q,R,\varepsilon )\). Thanks to the Holder’s inequality,
with Lemma 2.8, (2.63) and (2.62), we have for all \(s \geqq \max (-\log T, \tilde{S_3})\),
From now, we take \(\lambda > 2\) and we choose \(\varepsilon \in (0,\varepsilon _0]\) small enough. Observe that \(\lambda '> \frac{p+1}{p}\) since \(\lambda <p+1\). Let us now bound \(\Vert \psi \partial _s w \Vert _{L^{\lambda '}({\mathbf {B}}_{2R})}\). By using Holder’s inequality, we have
\(p_1=\frac{p+1}{p}\) and where
Putting (2.64) and (2.65) together, we get for all \(s \geqq \max (-\log T, \tilde{S_3})\),
By integrating inequality (2.67) between s and \(s+1\), we obtain for all \(s \geqq \max (-\log T, \tilde{S_3})\),
for some \(\tilde{q} > q\). Let \(\alpha = \frac{2}{(1 - \theta )\tilde{q}}\) and use Holder’s inequality in time, we obtain for all \(s \geqq \max (-\log T, \tilde{S_3})\),
From the inequalities (2.40), (2.68) and (2.69), we infer that for all \(s \geqq \max (-\log T, \tilde{S_3} )\),
Equipped with the arguments presented in the proof of Lemmas 6.5 and 6.6 in [9] and by exploiting Corollary 3, it is straightforward to get, for all \(s \geqq \max (-\log T, \tilde{S_3})\),
By combining (2.71), (B.7) and the identity \( e^{-\frac{ps}{p-1}}s^{\frac{a}{p-1}} |f(\phi w)|=\frac{1}{s^{a}}|w|^{p}\log ^a(2+\phi ^2w^2),\) we deduce that for all \(s \geqq \max (-\log T, \tilde{S_3})\),
where \({{\tilde{\varepsilon }}}=\frac{p(p-1)\varepsilon }{p+1-\varepsilon p}.\) Therefore,
Using together (2.51) and (2.73), we obtain
By Proposition 6.4 in [9], we have \( \frac{2p\tilde{q}\theta \alpha ' }{p+1-\varepsilon _1p} < 2q\), (for \(\varepsilon _1\) small enough) for all \(\tilde{q} \in [q, q + \frac{2}{p+1}]\). Then, by using the inequality, for all \(r\in [1, 2q]\), \(X^{r}\leqq C +CX^{2q}\), for all \(X>0\), we write for all \(s \geqq \max (-\log T, \tilde{S_3})\), for all \(\varepsilon \in (0,\varepsilon _1],\)
From (2.70) and (2.74), we have for all \(s \geqq \max (-\log T, \tilde{S_3})\),
Therefore, estimates (2.60) and (2.75) lead to the following:
Thus, inequality (2.54) is valid for all \(\tilde{q} \in [q, q + \frac{2}{p+1}]\). Repeating this argument, we would obtain that (2.54) holds for all \(q \geqq 2\). This concludes the proof of Proposition 2.5. \(\square \)
2.4 A Polynomial Bound for the \(H^{1}({\mathbf {B}}_R)\) Norm of Solution of Equation (1.18)
Based on Proposition 2.5, we are in position to derive a polynomial bound for the \(H^{1}({\mathbf {B}}_R)\) norm. More precisely, the aim of this subsection is to prove Proposition 2.1,
Proof of Proposition. 2.1
First, we use (2.40), Proposition 2.5 and apply Lemma A.1 with \(\alpha = q(p- \frac{ \varepsilon }{2}+1)\), \(\beta =p- \frac{ \varepsilon }{2}+1\), \(\gamma = \delta = 2\) to get that, for all \(s \geqq \max (-\log T, \tilde{S_3})\),
where \(\tilde{S_3}=\max (S_3,S_5)\). Clearly, there exists \(\varepsilon _2=\varepsilon _2(p,N,q)>0\) such that, for all \(\varepsilon \in (0,\varepsilon _2]\), we have \(q=\frac{2p- \varepsilon }{ \varepsilon }-1\geqq 2\). Therefore, for all \(\varepsilon \in (0,,\varepsilon _2],\) for all \(s \geqq \max (-\log T,{\tilde{S}}_3)\) we have
\(\square \)
We are now ready to Control of \(\nabla w\) in \(L^{2}({{\mathbf {B}}_R)}\). In fact, we use the Gagliardo-Nirenberg inequality in order to claim the following:
Lemma 2.9
There exists \(\varepsilon _3=\varepsilon _3(p,N)\in (0,\varepsilon _2]\) such that, for all \(\varepsilon \in (0,\varepsilon _3]\), for all \(s \geqq \max (-\log T, \tilde{S_3})\) we have
where \(\beta =\beta (p,N,\varepsilon ) \in (0,1)\) and \(\mu _{12}=\mu _{12}(R,\varepsilon )>0.\)
Proof
Let \(\varepsilon \in (0,\varepsilon _2)\). By interpolation, we write
where
where
and where \(\varepsilon <r-p-1\). Exploiting the fact that there exists \({{\tilde{\varepsilon }}}_2={{\tilde{\varepsilon }}}_2(p,N)\in (0,r-p-1)\) small enough such that for \(\varepsilon \in (0,{{\tilde{\varepsilon }}}_2],\) we have \(\nu =\nu (p,\varepsilon ) \in [1,2).\) Therefore, by using the properties of \(\psi \) given by (2.49) and the estimate (2.78) we get
Thanks to (2.80), (2.82) and the Sobolev embedding, we conclude
where
Note that, by exploiting the inequality \( |\nabla \big (\psi w\big )|^2\leqq 2 \psi ^2|\nabla w|^2 + 2|\nabla \psi |^2 w^2\), the properties of \(\psi \) given by (2.49) and the fact that \(\Vert \nabla \psi \Vert _{L^\infty }\leqq C\), we obtain
From (2.83), (2.84)and (2.42), we conclude
Now, if \(\varepsilon _3\leqq {{\tilde{\varepsilon }}}_2\) is chosen small enough such that \(\beta =\frac{r\varepsilon _3}{r-(p+1-\varepsilon _3)}\in (0,1)\), the estimate (2.85) implies (2.79). This ends the proof of Lemma 2.9.
Proof of Proposition 2.1:
From (2.50), the definition (2.48) of the local functional: \({\mathscr {E}}_\psi (w(s),s)\), we see that for all \(s \geqq \max (-\log T, S_4),\)
Thanks to (B.9) and (2.86) and the fact that \(\rho (2R)\leqq \rho (y)\leqq 1,\) for all \(y\in B_{2R} \), we conclude for all \(s \geqq \max (-\log T, S_4)\)
According to (2.87) together with Lemma 2.9 in the particular case when \(\varepsilon =\varepsilon _3\), we have for all \(s \geqq \max (-\log T, \tilde{S_3})\)
where \(\beta =\beta (p,N,\varepsilon _3) \in (0,1).\) It suffices to combine (2.88) and the fact that \(\beta <1\), to obtain that for all \(s \geqq \max (-\log T, \tilde{S_3})\)
Clearly, by combining (2.89), (2.42) and (2.49), we conclude (2.1), where \(\mu =\frac{\mu _{12}(R,\varepsilon _3)}{2-2\beta }\), which yields the conclusion of Proposition 2.1. \(\square \)
3 Proof of Theorems 1 and 2
In this section, thanks to polynomial estimate obtained in Proposition 2.1, we prove Theorems 1 and 2 here. This section is divided into two parts:
-
In subsection 3.1, we prove Theorem 1. More precisely, based upon Proposition 2.1, we construct a Lyapunov functional for equation (1.18) and a blow-up criterion involving this functional.
3.1 A Lyapunov Functional
In this subsection, our aim is to construct a Lyapunov functional for equation (1.18). Note that this functional is far from being trivial and makes our main contribution. More precisely, thanks to the rough estimate obtained in the Proposition 2.1, we derive here that the functional L(w(s), s) defined in (1.26) is a decreasing functional of time for equation (1.18), provided that is s large enough.
Let us remark that in Section 2, we construct a Lyapunov functional \(N_{m_0}(w(s),s)\) defined in (2.24), but we obtain just a rough estimate because the multiplier is not bounded. Nevertheless, the multiplier related to the functional L(w(s), s) is bounded. Then, as we said above, the natural energy E(w(s), s) defined in (2.40) is a small perturbation of L(w(s), s).
In order to prove that the functional L(w(s), s) is a Lyapunov functional, we start by using the additional information obtained in Section 2, to write several useful lemmas which play key roles in our analysis. More precisely, we start by stating the following:
Lemma 3.1
For all \(r\in [2,2^*)\), for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), we have
where \(\sigma =\mu (a,p,N,\frac{1}{2})\), \(M_1\) depends on p, a, N, r and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}\) and where \(2^*=\frac{2N}{N-2}\), if \(N\geqq 3\) and \(2^*=\infty \), if \(N=2\).
Throughout the proof we employ the following notations:
The ball in \({\mathbb {R}}^N\) with radius R around the point z is denoted \({{\mathbf {D}}(z,R)}=\{ x\in {\mathbb {R}}^N, \Vert x-z\Vert _{\infty }\leqq R \}\), where the infinity norm is given by the formula \(\displaystyle {\Vert x\Vert _{\infty }=\sup _{1\leqq i\leqq N}|x_i|}.\) Also, the ball in \({\mathbb {R}}^N\) with radius R around the point z is denoted \({{\mathbf {B}}(z,R)}=\{ x\in {\mathbb {R}}^N, |x-z|\leqq R \}\), where the norm is given by \(\displaystyle {|x|=\sqrt{\sum _{i=1}^Nx_i^2}}.\) Finally, let us recall that theses norms on \({\mathbb {R}}^N\) are equivalent. In fact, we have
Proof
In order to obtain the estimate (3.1), we combine a covering technique and the result obtained in Proposition 2.1.
First, we claim that \({\mathbb {R}}^N=\cup _{z\in Z^N} {{\mathbf {D}}(z,\frac{1}{2})}\) and the sequence \( \Big ({{\mathbf {D}}(z,\frac{1}{2})}\Big )_{z\in Z^N}\) are arbitrary pairwise sets are negligible. Let \(r\in [2,2^*]\). As an immediate consequence, we write
Note that using the definition (1.17) of \(w_{x_0},\) we see that
From (3.2) and (3.4), for all \( z \in {\mathbb {R}}^N\), \(s\geqq {\widehat{s}}_1=\max (-\log T,,{\widehat{S}}_1)\)
Thanks to (2.2) and (3.5), we have for all \( z \in {\mathbb {R}}^N\), \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\)
where \(\sigma =\mu (a,p,N,\frac{1}{2})\) and where \(M_2\) depends on p, a, N and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}\). By exploiting (3.6) and (3.3), we have for all \(x_0, z \in {\mathbb {R}}^N\), \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\)
To complete the proof, it remains to control the right-hand side of (3.7). More precisely, the term \(\displaystyle {\sum _{z\in Z^N} \sup _{y\in {\mathbf {D}}(z,\frac{1}{2})} \rho (y)}\). Using the fact that for all \(z\in {\mathbb {R}}^N\), for all \(y\in {{\mathbf {D}}(z,\frac{1}{2})}\), we have
Therefore, by using the basic inequality \((a+b)^2\leqq 2 a^2+2b^2,\) for all \(a,b >0,\) we set
In view of (3.9), (3.2), we have, for all \(z\in {\mathbb {R}}^N\), for all \(y\in {{\mathbf {D}}(z,\frac{1}{2})}\), we have
Due to (3.10) and to the definition of \(\rho \) given by (1.19), we conclude for all \(z\in {\mathbb {R}}^N\),
Thank to (3.11), we get
By combining (3.12) and (3.7), we easily obtain (3.1). This concludes the proof of Lemma 3.1.
Thanks to of Lemma 3.1, we are in position to state the following:
Lemma 3.2
For all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), we have
where, \(M_3\) depends on p, a, N and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}.\)
Remark 3.1
Let us mention that, in the first term on the right-hand side the choice of the power \(\frac{1}{4}\) is not optimal. In fact, with the same proof, one can show the same estimate with the power \(\nu \), for any \(\nu >0\), instead of the power \(\frac{1}{4}\). Let us denote that, we can construct a Lyapunov functional, when we have the estimate above for some power \(\nu \) such that \(\nu \in (0,1 )\) instead of the power \(\frac{1}{4}\).
Proof
Let \(\varepsilon \in (0,1)\). By using the inequality \(\log (2 +z^2)\leqq C+|z|^{\varepsilon ^2}\), for all \( z\in {\mathbb {R}}\), we conclude that
Furthermore, we apply the interpolation in Lebesgue spaces to get
By combining(B.4), (B.9) and the inequality \(|z|^{p+1+\varepsilon }\leqq 1+|z|^{p+1+2\varepsilon }\), for all \( z\in {\mathbb {R}}\), we obtain
Since \( p<p_S=\frac{N+2}{N-2}\), we then choose \( \varepsilon _4\) small enough, such that for all \(\varepsilon \in (0,\varepsilon _4 ]\) we have \(p+1+2 \varepsilon < 2^* \) where \(2^*=\frac{2N}{N-2}\), if \(N\geqq 3\) and \(2^*=\infty \), if \(N=2\). Therefore, estimate (3.1) implies that, for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), for all \(\varepsilon \in [0,\varepsilon _4]\),
where \(\sigma _3\) depends on \( p, a, N,\varepsilon _4\) and \(M_4\) depends on \( p, a, N,\varepsilon _4\) and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}\).
By combining (3.15), (3.16) and (3.17), we deduce that, for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), for all \( \varepsilon \in ( 0,\varepsilon _4]\).
Thanks to the basic inequality \(|a_1|^{\nu }|a_2|^{1-\nu }\leqq C|a_1|+C |a_2|\), for all \( a_1,a_2\in {\mathbb {R}}\), for all \( \nu \in ( 0, 1 ), \) we conclude that, for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), for all \( \varepsilon \in ( 0,\varepsilon _4]\),
Now, we choose \( \varepsilon _5\in ( 0,\varepsilon _4], \) such that \( \sigma _3\varepsilon _5 <\frac{1}{4}.\) Then, by (3.14) and (3.19), we easily obtain (3.13). This concludes the proof of Lemma 3.2.
Thanks to estimate (3.13), we can improve the estimate (2.8) related to the control of the time derivative of the functional E(w(s), s). More precisely, we prove the following lemma:
Lemma 3.3
There exists \({\widehat{S}}_2>{\widehat{S}}_1\) such that for all \(s\geqq {\widehat{s}}_2=\max (-\log T, {\widehat{S}}_2)\), we have
where, \(M_7\) depends on p, a, N and \(\Vert w( {\widehat{s}}_1)\Vert _{H^1}.\)
Proof
By using the additional information obtained in (3.13), we are going to refine the estimate related to \(\Sigma _{1}^{2}(s)\) and \(\Sigma _{1}^{3}(s)\) defined in (2.10). Let us mention that the estimate (2.12) related to \(\Sigma _{1}^{1}(s)\) defined in (2.10) is acceptable and does not need any improvement. More precisely, we write
We attempt to group the main terms together. A straightforward computations implies that
where
where \(F_1\) and \(F_2\) are defined by (2.13) and (2.14).
Note that, in (3.21) we grouped the main terms together. In fact, it is easy to control the terms \(\chi _{2}(s)\). However, the control of the term \(\chi _{1}(s)\) needs the use of the additional information obtained in Lemma 3.2. More precisely, for all \(s\geqq {\widehat{s}}_1=\max (-\log T,{\widehat{S}}_1)\), we divide \({\mathbb {R}}^N\) into two parts
Accordingly, we write \(\chi _1(s)=\chi _1^1(s)+\chi _1^2(s)\), where
On the one hand, by using the definition of the set \(A_1(s)\) given in (3.24), we get, for all \(s\geqq {\widehat{s}}_1\),
From (3.25) and the fact that \(1-\frac{2s}{(p-1)\log (2+\phi ^2 w^2)} \leqq 1\), we get
On the other hand, by using the definition of the \(\phi (s)\) given by (1.20), we write the identity
Now, by using the inequality \(\phi (s)\geqq 1 \) and (3.27), we write for all for all \(s\geqq {\widehat{s}}_1\),
Also, by using the definition of the set \(A_2(s)\) defined in (3.24), we can write for all \(s\geqq {\widehat{s}}_1\), if \(y\in A_{2}(s)\), we have
Clearly, the exists \(S_2>S_1 \) such that for all \(s \geqq S_2\), we have \( \frac{2s}{p-1}-\frac{a \log s}{p-1}\geqq \frac{s}{p-1}\). Therefore, by exploiting (3.28) and (3.29) we have for all \(s\geqq {\widehat{s}}_2=\max (-\log T, {\widehat{S}}_2)\),
Note that, by using the fact \(\chi _1(s)=\chi _1^{1}(s)+\chi _1^{2}(s)\), (3.13), (3.26) and (3.30), we get for all \(s\geqq {\widehat{s}}_2=\max (-\log T, {\widehat{S}}_2)\),
Thanks to (B.5) and (B.6), we write
By (2.10), (3.32) and (2.17), we have, for all \(s\geqq {\widehat{s}}_1\),
The result (3.20) derives immediately from (2.10), (2.12), (3.31), (3.33), and the identity (3.21), which ends the proof of Lemma 3.3
With Lemmas 2.3 and 3.3, we are in a position to prove Theorem 1.
Proof of Theorem 1:
By exploiting the defintion of \(L_0(w(s),s)\) in (2.4), we can write easily, for all \(s\geqq {\widehat{s}}_2=\max (-\log T,{\widehat{S}}_2)\),
where \(J(w(s),s)=\frac{1}{s}\int _{{\mathbb {R}}^N}w^2\rho (y){\mathrm {d}}y\). Lemmas 2.3 and 3.3 allows to prove that for all \(s\geqq {\widehat{s}}_2=\max (-\log T,{\widehat{S}}_2)\), we have
Again, choosing \({\widehat{S}}_3> {\widehat{S}}_2\) large enough, this implies that for all for all \(s\geqq \max (-\log T,{\widehat{S}}_3)\), we have
Recalling that,
we get from straightforward computations
Therefore, estimates (3.35) and (3.36) lead to the following crucial estimate:
Since we have \(1\leqq \exp \Big (\frac{p+3}{\sqrt{s}}\Big )\leqq \exp \Big (\frac{p+3}{\sqrt{ {\widehat{S}}_3}}\Big )\), we then choose \(\theta \) large enough, so that \( M_9\exp \Big (\frac{p+3}{\sqrt{s}}\Big )-\frac{4\theta }{3} \leqq 0\), which yields, for all \(s\geqq s_3=\max (-\log T,{\widehat{S}}_3)\),
A simple integration between s and \(s+1\) ensures the result. This concludes the proof of Theorem 1. \(\square \)
We now claim the following lemma:
Lemma 3.4
There exist \(M_{10}>0\) and \({\hat{S}}_{4}\geqq {\hat{S}}_3\) such that, we have for all \(s\geqq \max ( \hat{S}_4,-\log T)\)
Proof
The argument is the same as the similar part in Proposition 2.4.
3.2 Proof of Theorem 2
As in [9], by combining Theorem 1 and Lemma 3.4 we get the following bounds:
Corollary 4
For all \(s\geqq \max (-\log T,{\widehat{S}}_4)\), we have
where \(M_{11}, M_{12},M_{13},... M_{19} \) depend on \( p, a, N, s_3 = \max (-\log T,{\widehat{S}}_3)\) and \(\Vert w( s_3)\Vert _{H^1}\).
Let us denote that, the estimates obtained in the above corollary are similar to the Corollary (3) except for the presence of the term \(K_is^{b+1}\) instead of \(M_i\). Consequently, following the proof of Proposition 2.5 line by line we are in position to prove the following:
Proposition 3.5
For all \(q \geqq 2\), \(\varepsilon >0\) and \(R>0\) there exist \(\varepsilon _6=\varepsilon _6(q,R)>0\), there exists a time \({\widehat{S}}_5(q,R,\varepsilon )\geqq {\widehat{S}}_4\), such that for all \(s\geqq \max (-\log T,{\widehat{S}}_5)\), we have
where \(M_{20}(q,R,\varepsilon )\) depends on \( p, a, N,q, R,\varepsilon , s_3 = \max (-\log T,{\widehat{S}}_3)\) and \(\Vert w( s_3)\Vert _{H^1}\).
Finally, we are in position to prove Theorem 2 by exploiting Lemma A.1 and Lemma A.2.
Proof of Theorem 2
First, we use (3.40), Proposition 3.5 and apply Lemma A.1 with \(\alpha = q(p- \frac{ \varepsilon }{2}+1)\), \(\beta =p- \frac{ \varepsilon }{2}+1\), \(\gamma = \delta = 2\) to get that, for all \(s \geqq \max (-\log T,{\widehat{S}}_5)\),
Hence, for all \(\varepsilon \in (0,p-1)\), we have \(q=\frac{4p-4-\varepsilon }{\varepsilon }\geqq 2\). Therefore, the estimate (2.77) implies
Let us recall the equation in w:
where \(\phi (s)\) and f are given in (1.20) and (1.2).
We now apply Lemma A.2 to w, with \(b =b(y)= \frac{1}{2}y\) and
From (B.7), we see that, for all \(\varepsilon \in (0,p-1)\), we have
Let \(\lambda _1= p+1-\varepsilon \), \(\alpha _1=\frac{\lambda _1}{p-1+\varepsilon }\) and \(\beta _1=\frac{1}{\varepsilon }\). Thus, the first identity in (A.2) holds with \(g(y,s,w) = C(\varepsilon )(|w(y,s)|^{p-1+\varepsilon } + 1)\). Since \(p < \frac{N + 2}{N - 2}\), then we can choose \(\varepsilon _7\leqq \varepsilon _6\) small enough, such that the conditions \(\frac{1}{\beta _1} + \frac{N}{2\alpha _1} < 1\) and \(\alpha _1\geqq 1\) hold. Moreover, for all \(s \geqq \max (-\log T ,{\widehat{S}}_5)\) we have
By exploiting (3.51) and (3.49), we deduce that
Then the second condition in (A.2) holds. Therefore,
for some \(\tau _0 \in (0,1)\). By (3.53), we write
for some \(\tau _0 \in (0,1).\) From the fact that the above estimate is independent of \(x_0\) and the definition of \(w_{x_0}\) given by (1.17), we infer
This concludes the proof of Theorem 2. \(\square \)
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Appendices
Appendix
We recall the interpolation result from Cazenave and Lions [1] and the interior regularity theorem in [6].
Lemma A.1
(Interpolation technique, Cazenave and Lions [1]) Let \(t_0>0\). Assume that
for some \(1< \alpha , \beta , \gamma , \delta < \infty \). Then
for all \(\lambda < \lambda _0 = \frac{(\alpha + \gamma ')\beta \delta }{\gamma '\beta + \alpha \delta }\) with \(\gamma ' = \frac{\gamma }{\gamma - 1}\), and satisfies
for \(\lambda < \lambda _0\). The positive constant C depends only on \(\alpha , \beta , \gamma , \delta , N\) and R.
The second one is an interior regularity result for a nonlinear parabolic equation:
Lemma A.2
(Interior regularity) Let \(v(x,t)\in L^\infty \big ((0,+\infty ), L^2({\mathbf {B}}_R)\big ) \cap L^2\big ((0,+\infty ), H^1({\mathbf {B}}_R)\big )\) which satisfies
where \(R > 0\), \(|b(x,t)| \leqq \mu _1\) in \(Q_R\) and \(|H(x,t,v)| \leqq g(x,t)(|v| + 1)\) with
and \(\frac{1}{\beta '} + \frac{N}{2\alpha '} < 1\), and \(\alpha ' \geqq 1\). If
and \(\mu _1\), \(\mu _2\) and \(\mu _3\) are uniformly bounded in t, then there exists a positive constant C depending only on \(\mu _1\), \(\mu _2\), \(\mu _3\), \(\alpha '\), \(\beta '\), N, R and \(\tau \in (0,1)\) such that
Some Elementary Lemmas
Let f, F, \(F_2\) be the functions defined in (1.2), (1.25) and (2.14). Clearly, we have
Lemma B.1
Let \(q>1\),
Proof
See Lemma A.1 in [15]. \(\square \)
Thanks to (B.1), (B.2) and (B.3), we will give the first and the second order terms in the expansion of the nonlinearity F(x) defined in (1.25), when |x| is large enough. More precisely, we now state the following estimates:
Lemma B.2
For all \(s \geqq 1\), for all \(z\in {\mathbb {R}}\),
where \(\phi \), F, \(F_1\) and \(F_2\) are given in (1.20), (1.25), (2.13) and (2.14).
Proof
Note that (B.4) obviously follows from (B.2). In order to derive estimates (B.5) and (B.6), considering the first case \(z^2\phi (s)\geqq 4\), then the case \(z^2\phi (s)\leqq 4\), we would obtain (B.5) and (B.6) by using (B.1), (B.2) and(B.3). Similarly, by taking into account the inequality \(\log ^a (2+u^2)\leqq C(\varepsilon )+ C(\varepsilon ) |u|^{\varepsilon }\) , we conclude easily (B.7), (B.8), (B.9) and (B.10). This ends the proof of Lemma B.2. \(\square \)
Proof of Proposition 2.6
Let us first derive the upper bound for \({\mathscr {E}}_\psi \).
Proof
(Proof of the upper bound for \({\mathscr {E}}_\psi \)) Multiplying (1.18) by \( \partial _{s} w \psi ^2\rho (y)\) and integrating over \({\mathbb {R}}^N,\) we obtain
Proceeding similarly as for the terms \(\Sigma _{1}^{1}(s)\), \(\Sigma _{1}^{2}(s)\) and \( \Sigma _{1}^{3}(s)\) defined in (2.10), we get
Using the fact that \(2ab \leqq \frac{a^2}{4} + 4b^2\), we obtain
which implies, for all \(s\geqq \max (-\log T,1)\),
where \(C = C(a, p, N, \Vert \psi \Vert _{L^\infty }, \Vert \nabla \psi \Vert _{L^\infty })\).
By combining (C.3), (2.40) and (2.41), we infer for all \(s\geqq \max (-\log T,S_2)\)
From the definition of \({\mathscr {E}}_\psi \) given in (2.48), using the fact that, \(F(\phi w)\geqq 0,\) we have
By the definition of \(H_{m}(w(s),s)\) given in (2.5), exploiting (2.39), we write for all \(s\geqq \max (-\log T,S_2)\)
Integrating the inequality (C.5) from s to \(s+1\) and using (2.17), (B.4) and (2.41) we get, for all \(s\geqq \max (-\log T,S_2)\)
By using the mean value theorem, we derive the existence of \(\sigma (s)\in [s,s+1]\) such that
Let us write the identity, for all \(s\geqq \max (-\log T,S_2)\)
By combining (C.6), (C.7) and (C.4), we infer, for all \(s\geqq \max (-\log T,S_2)\)
This concludes the proof of the upper bound for \({\mathscr {E}}_\psi \). \(\square \)
It remains to prove the lower bound.
[Proof of the lower bound for \({\mathscr {E}}_\psi \)]
Consider now, for all \(s \geqq \max (-\log T,1)\),
Multiplying equation (1.18) with \(\psi ^2 w,\) integrating on \(\mathbb {R}^N\) and using the same argument as in the proof of Lemma 2.3 yields
Therefore, there exists \({\tilde{S}}_2>S_2\) large enough, such that for all \(s \geqq \max (-\log T, {\tilde{S}}_2)\), we have
Furthermore, after some integration by parts, we write
Thanks to the estimates \( \Vert \psi \Vert ^2_{L^\infty } +\Vert \Delta \psi \Vert ^2_{L^\infty } + \Vert \nabla \psi \Vert ^2_{L^\infty }+ \Vert y.\nabla \psi \Vert ^2_{L^\infty }\leqq C\), (C.11) and (2.42), we have for all \(s \geqq \max (-\log T, {\tilde{S}}_2)\),
Using (C.10) and (C.12), we obtain for all \(s \geqq \max (-\log T, {\tilde{S}}_2),\)
Let us define the following functional:
where \({\mathscr {G}}_{\psi }(w(s),s) \) is defined in (2.48).
We claim that the function of \({\mathscr {G}}_{\psi }(w(s),s) \) is bounded from below by some constant M, where M is a sufficiently large constant that will be determined later. Arguing by contradiction, we suppose that there exists a time \(s^* \geqq \max (-\log T, {\tilde{S}}_2)\) such that \({\mathscr {G}}_{\psi }(w(s^*),s^*) \leqq - Q\), for some \(Q>0\). Then, we write
If we now compute the time derivative of \({\mathscr {G}}_{\psi }(w(s),s) \) we get for all \(s \geqq s^*,\)
From the definition of \({\mathscr {E}}_\psi \) given in (2.48), using (B.4) and (2.17) we have for all \(s \geqq s^*,\)
Thanks to (C.4) we conclude for all \(s \geqq s^*,\)
Moreover, from (2.41), we obtain for all \(s \geqq s^*,\)
Integrating the identity (C.16) over \([s^*,s]\) and combining (C.17), (C.18) and (C.19) we deduce that
Combining (C.13), (C.15) and (C.20) we infer for all \(s\geqq s^*,\)
Thanks to (B.4) and (B.10), we have for all \(s\geqq s^*,\) that
Due to Jensen inequality, (C.21) and (C.22) we find for all \(s\geqq s^*,\)
where \({\tilde{Q}}=Q-C_5\).
It is interesting to denote that we easily prove that the solution of the differential inequality
blows up in finite time before
Now, we choose \(Q= Q_{9}T^* +C_5+ 1\) to get \({\tilde{Q}} - Q_{9}(s-s^*) \geqq 1\) for all \(s \in [s^*, s^* + T^*]\).
Therefore, \({\mathscr {I}}_\psi (w(s),s)\) blows up in some finite time before \(s^* + T^*\). But this contradicts with the global existence of w. This implies (2.50), and we complete the proof of Proposition 2.6. \(\square \)
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Hamza, M.A., Zaag, H. The Blow-Up Rate for a Non-Scaling Invariant Semilinear Heat Equation. Arch Rational Mech Anal 244, 87–125 (2022). https://doi.org/10.1007/s00205-022-01760-w
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DOI: https://doi.org/10.1007/s00205-022-01760-w