Abstract
We study the general family of nonlinear evolution equations of fractional diffusive type \(\partial _t u-\text{ div }\left( |u|^{m_1}\nabla (-\Delta )^{-s} [ |u|^{m_2-1}u]\right) =f\). Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider \(m_1, m_2>0\), and \(s\in (0,1)\), and prove the existence of weak solutions. Moreover, when \(f\equiv 0\) we obtain the \(L^p\)-\(L^\infty \) decay estimates of solutions, for \(p\geqq 1\). In addition, we also investigate the finite time extinction of solution.
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1 Introduction
The main purpose of this paper is to study the following evolution equation of diffusive type with nonlocal effects:
with \(m_1, m_2>0\), \(s\in (0,1)\), and space dimension \(N\geqq 2\). The symbol \( (-\Delta )^{-s}\) denotes by the inverse of the fractional Laplacian operator as usual (see, for example [28]).
Equation (1.1) corresponds to the well-known Porous Medium Equation \(\partial _t u = \text{ div } ( u^{m_1} \nabla u)\), when one considers \(s=0\), and \(m_2=1\). This model arises from considering a compressible fluid with a density distribution u(x, t) and with Darcy’s law leading to the equation
where p denotes the pressure. Many other different relations between the density, the velocity and the pressure arise in the applications. For example, the model proposed by Leibenzon and Muskat states a law in which \(p=g(u)\), where g is a nondecreasing scalar function (see more examples in [31]). Equations of this type have been studied by many authors (see, for example [3, 5,6,7,8,9,10,11,12,13,14, 17,18,19, 21, 26, 29, 32]). There are many questions, addressed to equations of this type, which are the object of active research, such as those pertaining to existence and uniqueness, regularity, the behaviour of solutions in short time and in large time, the finite and infinite speed of propagation, and so on.
Here, we would like to mention recent results, close to the ones in our paper. Equation (1.1) with \(m_1=m_2=1\), which reads as: \(u_t = \text{ div } (u \nabla (-\Delta )^{-s} u)\) was first introduced by Caffarelli and Vázquez [11]. In [3], Biler et al. studied a particular case of equation (1.1):
where \(\alpha =2(1-s)\in (0,2), m_1=1, m=m_2+1\), and \(f=0\). The authors constructed nonnegative self-similar solutions; the so called Barenblatt–Pattle–Zeldovich solutions. Furthermore, they proved the existence of weak solutions for \(u_0\in L^1(\mathbb {R}^N) \cap L^\infty ( \mathbb {R}^N)\), and the decay estimate \(L^1\)-\(L^p\) (see Theorem 1) as follows:
Thanks to this decay, they also obtained the existence of a solution for \( u_0\in L^1(\mathbb {R}^N)\), under the assumptions
Equation (1.1) with \(s\in (0,1), m_2=1, m_1=m-1>0\), and \( f=0\) was investigated by Stan et al. in [27]. The authors studied the existence of nonnegative weak solutions for all integrable initial data \( u_0\). In addition, they obtained the smoothing effect \(L^p\)-\(L^\infty \) for \( p\geqq 1\):
with \(C=C(N,s,m,p)>0\). In particular, by considering the case of \(p=1\), (1.3) allowed them to obtain the existence result for initial data with bounded measure. Moreover, the finite and infinite speed of propagation were also studied by the same authors; see [26] (see also [4] for a different equation of this type).
It is also interesting to note that the mean field equation
could be considered as a limit of (1.1) with \(m_1=m_2=1\), and \(f=0\), as \(s\rightarrow 1^{-}\). In fact, Serfaty and Vázquez [24] proved the existence of a solution of (1.4) for all integrable initial data, even for data measure. A uniqueness result was also given in the class of bounded solutions. Furthermore, the solution, constructed in [24], satisfies a universal bound
with \(C=C(N)>0\).
Very recently, Nguyen and Vázquez [21] proved the existence of weak solutions of (1.1) in a bounded domain \(\Omega \subset \mathbb {R}^N\), with the homogeneous Dirichlet boundary condition. Furthermore, they also obtained a universal bound
with \(m_1+m_2>1\) and \(C=C(N,s,|\Omega |, m_1+m_2)\).
The main goal of this paper is to carry out a qualitative study of weak solutions of (1.1). We first prove the existence of weak solutions with data \(u_0\in L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N)\), and \(f\in L^1(Q_T)\cap L^\infty (Q_T)\), where \(Q_T=\mathbb {R}^N\times (0,T)\). Moreover, when \(f=0\) we show \(L^p\)-\(L^\infty \) decay estimates of solutions for all \(p\geqq 1\); see Theorem 2 below. We also emphasize that our decay results below hold for \(m_1+m_2> 1-\frac{2p(1-s)}{N}\). Thus, we improve upon the previous range of \(m=m_1+m_2>1\), described in (1.2) and (1.3). For the case \(m_1+m_2<1-\frac{2p(1-s)}{N}\) and \(f=0\), we show that every weak solution vanishes in a finite time; see Theorem below. In addition, we also obtain the regularity of
for any \(R>0\), provided that either \(s\in [\frac{1}{2}, 1)\) or \(m_2> m_1\). Also,
if \(s\in (0,\frac{1}{2})\); see Propositions 8 and 9 below. These improve the regularity \(\text{ div }\Theta (u)\in L^1 (0,T, (W^{2,\infty }_0(B_R))^\prime )\) of Nguyen and Vázquez [21].
Our proof is self contained, and it is merely based on the Fourier analysis and the fundamental estimates of the Riesz potential. This enables us to avoid using the spectral theory approach, which is useful in studying equations of this type on a bounded domain with the homogeneous boundary condition (see for example [6, 21, 27]), or to avoid using the characterization of Besov and Triebel–Lizorkin space in order to obtain some estimates involving the fractional Sobolev spaces \(W^{s, p}\), see for example [3].
1.1 Definition and main results
Let us put \(\Theta (u)= |u|^{m_1}\nabla (-\Delta )^{-s} [|u|^{m_2-1}u]\). Now, we introduce first the definition of a weak solution that we are going to use in this paper.
Definition 1
Let \(u_0\in L^1(\mathbb {R}^N) \cap L^\infty ( \mathbb {R}^N)\) and \(f\in L^1(Q_T) \cap L^\infty (Q_T)\). We say that u is a weak solution of problem (1.1) if \(u\in L^1(Q_T) \cap L^\infty (Q_T)\) satisfies \(\text{ div }\Theta (u)\in L^2\left( 0,T, Y(B_R)\right) \), and
where
Note that \(H^{-1}(B_R)\)\(\left( \text {resp. } W^{-2,p}(B_R)\right) \) is the dual space of \(H^1_0(B_R)\)\(\left( \text {resp. } W^{2,p}_0(B_R)\right) \), and \(B_R\) is the ball in \(\mathbb {R}^N\), with center at 0 and radius R.
Remark 1
It follows from the Definition 1 that \(u\in \mathcal {C} \left( [0,T]; Y(B_R)\right) \) for any \(R>0\). Thus, u(t) possesses an initial trace \(u_0\) in this sense. Particularly, if either \(s\in [\frac{1}{2} ,1)\) or \(m_2>m_1\), then \(u\in \mathcal {C}\left( [0,T]; H^{-1}(B_R)\right) \) for every \(R>0\).
Under this framework, our existence result is as follows:
Theorem 1
Let \(m_1, m_2>0\) and \(s\in (0,1)\). Let \(u_0\in L^1(\mathbb {R}^N) \cap L^\infty (\mathbb {R}^N)\) and \(f\in L^1(Q_T) \cap L^\infty (Q_T)\). Then, there exists a weak solution u of (1.1). Moreover, u satisfies the following properties:
-
(i) \(L^q\)-estimate For any \(1\leqq q \leqq \infty \), we have
$$\begin{aligned} \Vert u(t)\Vert _{L^q} \leqq \Vert u_0\Vert _{L^q}+ t^\frac{q-1}{q} \Vert f\Vert _{L^q(Q_t)}, \quad \text {for almost everywhere }\, t\in (0,T). \end{aligned}$$(1.5)Here, we denote \(\frac{q-1}{q}=1\) if \(q=\infty \).
-
(ii) Energy estimates
If \(m_2>m_1\), then there is a constant \(C=C(u_0, f, m_1, m_2)>0\) such that
$$\begin{aligned} \Vert (-\Delta )^{\frac{1-s}{2}} [|u|^{m_2-1} u] \Vert _{L^2(Q_T)}\leqq C. \end{aligned}$$(1.6)If \(m_2=m_1\), then there is a constant \(C=C(u_0, f, m_2)>0\) such that
$$\begin{aligned} \Vert (-\Delta )^{\frac{1-s}{2}} [|u|^{m_2 p_0-1} u] \Vert _{L^2(Q_T)}\leqq C, \end{aligned}$$(1.7)with \(p_0=\frac{N+2(1-s)}{N+2(1-2s)}\).
If \(m_2<m_1\), then there is a constant \(C=C(u_0, f, m_1, m_2)>0\) such that
$$\begin{aligned} \Vert (-\Delta )^{\frac{1-s}{2}} [|u|^{m_1 -1} u] \Vert _{L^2(Q_T)}\leqq C. \end{aligned}$$(1.8)
Next, we provide a sharper decay result of solution of (1.1) for the case \(f=0\).
Theorem 2
Let \(p \geqq 1\), and \(s\in (0,1)\). Let \(m_1, m_2>0\) be such that \(m_1+m_2> 1-\frac{2 p (1-s)}{N}\). Assume that \(f=0\) and \( u_0\in L^{p}(\mathbb {R}^N)\). Then, there exists a constant \(C=C(N,s,m_1+m_2, p)>0\) such that
with \(\alpha _0=\frac{N-2(1-s)}{N}\), and \(\beta _0=m_1+m_2-1\).
Remark 2
We emphasize that (1.9) holds for the case \(m_1+m_2> 1- \frac{2p(1-s)}{N}\). Thus, we improve the decay result of the authors in [3, 27], where the authors assumed \(m=m_1+m_2>1\).
Finally, we study the finite time extinction of solution.
Theorem 3
Let \(s\in (0,1)\), and \(m_1, m_2>0\) be such that \(m_1+m_2<\alpha _0\). Assume that \(f=0\) and \(u_0\in L^{1}(\mathbb {R} ^N)\cap L^{\infty }(\mathbb {R}^N)\). Then, there is a finite time \(\tau _0>0\) such that
Our paper is organized as follows: the next section is devoted to review the fractional Sobolev spaces, and the approximation of the fractional Laplacian \((-\Delta )^s\). Moreover, we prove some functional inequalities which will be useful later. In Section 3, we study the existence of a solution to a regularized equation to (1.1), and we justify passing to the limit in order to obtain the existence of a solution to (1.1). The last section is devoted to investigating some decay estimates, and the extinction in a finite time of weak solutions.
Throughout this paper, the constant C may change step by step. Moreover, \( C=C(\alpha , \beta , \gamma )\) means that the constant C merely depends on the parameters \(\alpha , \beta ,\gamma \). We denote \(\Vert .\Vert _{X(\mathbb {R}^N)}= \Vert .\Vert _X\), and \(\int _{\mathbb {R}^N} f(x)\hbox {d}x =\int f(x)\hbox {d}x\) for short. Finally, the notation \(A\lesssim B\) means that there exists a positive constant \(c>0\) which is independent of data such that \(A\leqq cB\).
2 Functional Setting
Let \(p\geqq 1\), and \(s\in (0,1)\). For a given domain \( \Omega \subset \mathbb {R}^N\), we define the fractional Sobolev space
endowed with the norm
Moreover, we also denote the homogeneous fractional Sobolev space by \(\dot{W} ^{s, p} (\Omega )\), endowed with the seminorm
In particular, we denote \(W^{s,2}(\mathbb {R}^N)\) by \(H^s(\mathbb {R}^N)\), which turns out to be a Hilbert space. It is well-known that we have the equivalent characterization
where \(\mathcal {F}\) denotes the Fourier transform, and that we have
In addition, for \(u\in H^{s}(\mathbb {R}^N)\), the fractional Laplacian is defined by
Then,
We emphasize that if \(s>0\), then \((-\Delta )^{-s}=\mathcal {I}_{2s}\), the Riesz potential, (see, for example [28]). Moreover, the fractional gradient \(\nabla ^{s}\) can be written as \(\nabla \mathcal {I}_{1-s}\). For any smooth bounded function \(v: \mathbb {R}^N\rightarrow \mathbb {R}\), we have
with a suitable constant C(N, s), see [3].
2.1 Approximation of the Fractional Laplacian \((-\Delta )^s\)
For fixed \(s\in (0,1)\), and each \(\varepsilon >0\), let us define the operator
for \(x\in \mathbb {R}^N\), and for \(f\in \mathcal {S}(\mathbb {R}^N)\) (the Schwartz space). It is known that the operator \(\mathcal {L}^s_\varepsilon \) can be considered as a regularization of the fractional Laplacian \( (-\Delta )^s\), see [9]. Next, we recall some properties of the operator \(\mathcal {L}^s_\varepsilon \).
-
Square root By the symmetry, we observe that
$$\begin{aligned} \left<\mathcal {L}^s_\varepsilon [f] , f\right>_{L^2} = \frac{C}{2} \int \int \frac{|f(x)-f(y)|^2}{ (|x-y|^2 + \varepsilon ^2)^{\frac{N+2s}{2}} } \hbox {d}x\hbox {d}y. \end{aligned}$$Then, we denote \(\mathcal {L}^{\frac{s}{2}}_\varepsilon [f]\) as a square root of \(\mathcal {L}^{s}_\varepsilon [f]\) in the Fourier transform sense, and
$$\begin{aligned} \Vert \mathcal {L}^{\frac{s}{2}}_\varepsilon [f]\Vert ^2_{L^2}= \left<\mathcal {L}^s_ \varepsilon [f] , f\right>_{L^2}. \end{aligned}$$
Lemma 1
Let \(f\in H^1(\mathbb {R}^N)\), and \(s\in (\frac{1}{2 },1)\). Then, it holds that
where the constant \(C=C(N,s)>0\).
Proof
It follows from the Plancherel theorem that
On the other hand, we have
We skip the proof of this inequality, and refer to Lemma 10. Thus, we obtain
By Hölder’s inequality, we have
From (2.4) and (2.5), we get the result. \(\quad \square \)
Lemma 2
Let \(\{f_\varepsilon \}_{\varepsilon >0}\) be a sequence in \(L^2(\mathbb {R}^N)\) such that \(f_\varepsilon \rightarrow f\) in \( L^2(\mathbb {R}^N)\) as \(\varepsilon \rightarrow 0\). Then, for any \(s\in (0,1)\), it holds that
Proof
From the triangle inequality, we have
By applying Lemma 10 in the “Appendix”, we have
Then, we obtain
Similarly, we also get
Moreover, we observe that \(\mathcal {F} \left\{ (-\Delta )^{-s} \mathcal {L}^s_\varepsilon [f] -f \right\} (\xi ) \rightarrow 0\), for every \(\xi \in \mathbb {R}^N\). Thanks to the Dominated Convergence Theorem, we conclude
as \(\varepsilon \rightarrow 0\).
A combination of (2.7), (2.8) and (2.9) yields the proof of Lemma 2. \(\quad \square \)
Next, we prove a generalized version of Stroock–Varopoulos’s inequality.
Lemma 3
(Generalized Stroock–Varopoulos Inequality for \(\mathcal {L}^s_ \varepsilon \)) Let \(s\in (0,1)\), and let \(\psi , \phi \in \mathcal {C}^1(\mathbb {R})\) be such that \(\psi ^\prime , \phi ^\prime \geqq 0\). Then,
If we take \(\psi (f)=f\), then we obtain
where \(\phi ^\prime =(\Phi ^\prime )^2\).
Proof
We have
Since \(\psi ^\prime , \phi ^\prime \geqq 0\), we have
Hence, we get (2.10).
Finally, (2.11) is proved in Theorem 3.2, [27]. \(\quad \square \)
To end this part, we point out some well-known fundamental inequalities, used several times in this paper.
Lemma 4
For any \(\alpha >0\) and \(\beta \in (0,1)\), it holds that
Lemma 5
Let \(\alpha , \beta >0\), and \(\theta =\frac{ \alpha +\beta }{2}\). Then, there is a constant \(C>0\) such that
3 A Regularized Problem
In this section, we study the solutions of the following problem:
where \(s_0=(1-2s)_+\), \(\Theta _{\varepsilon ,\nu } (u) = H_\nu (u) \nabla (-\Delta )^{-1} \mathcal {L}^{1-s}_\varepsilon [ G_\nu (u)]\), and
with \(m_0=\frac{1}{2}\displaystyle \min \{m_1, \frac{m_2(N-2s_0)}{N}\}\), and for every \(\delta _1, \delta _2, \varepsilon , \kappa , \nu \in (0,1)\). Note that (3.1) is a regularization of (1.1). We shall prove the existence of solutions of (3.1) in a suitable functional space by using the fixed-point theorem, and derive some energy estimates. Let us put
The associated norm \(\Vert .\Vert _X\) is \(\Vert .\Vert _{L^1(\mathbb {R}^N)}+ \Vert .\Vert _{L^{\infty }(\mathbb {R}^N)}\). Then, we have
Theorem 4
Let \(u_0\in X\) and \(f\in L^1(Q_T)\cap L^\infty (Q_T) \). Then, there exists a weak solution \(u\in \mathcal {C}([0,T]; X)\) satisfying problem (3.1) in the weak sense, that is,
for all \(\varphi \in \mathcal {C}^\infty _c (Q_T)\).
Proof
To prove Theorem 4, we first look for a mild solution \(u\in \mathcal {C}([0,T]; X)\) as a fixed point of the map
where \(e^{t\Delta }\) is the semigroup corresponding to the heat kernel \((4\pi t)^{-\frac{N}{2}} \exp (-\frac{|x|^2}{4t})\). Furthermore, we have a fundamental estimate for the heat semigroup \(e^{t \Delta }\) (see Proposition 1.2, Ch. 15, [30]).
Proposition 1
For every \(1\leqq q\leqq r\leqq \infty \), it holds that
where the constant \(C>0\) depends on the parameters involved.
Proof
The proof of Proposition 1 is quite easy: it follows from Young’s inequality for convolution, so we skip the detail and leave it to the reader. \(\quad \square \)
Next, the following lemma shows that the operator \(\mathcal {T}\) is a local contraction:
Lemma 6
For any \(T\in (0,1)\), the operator \(\mathcal {T}\) maps \(\mathcal {C}([0,T]; X)\) into itself. Moreover, there is a real number \(\gamma \in (0,1)\) such that for all \(u, v\in \overline{B}(0,R) \subset \mathcal {C}([0,T]; X)\),
where C(R) depends on R and the parameters involved.
Proof of Lemma 6
Let us drop the dependence on the parameters \(\varepsilon , \nu , \kappa \) of the terms \(\Theta _{\varepsilon ,\nu }, H_\nu , G_\nu , J_\kappa \) for short. Then, we have
By applying Proposition (1), we obtain
Obviously, we will consider \(r=1\) and \(r=\infty \) alternatively in what follows.
We now consider the first term on the right hand side of (3.4). Let us write
Let us fix \(q>\frac{N}{N-1}\), and put \(q'=\frac{q}{q-1}\), \(q^\star =\frac{Nq}{N+q}\). Then,
Thus,
Note that the second inequality is obtained by using the well known property of Riesz potential \(\mathcal {I}_{1}\), and the fourth inequality follows from the interpolation inequality that
Similarly, we also get
By choosing \(r=q=1\) in (3.4), and by (3.5), (3.6), we obtain
for any \(t\in (0,T)\), with \(C_2(R, \varepsilon )=\max \{C_1(R), C(R,\varepsilon )\}\).
Next, we estimate \(\Vert A\Vert _{L^q}\) for every \(q>N\). In a similar way to the proof of (3.5), we have
By the same argument as in (3.6), we obtain
Now, let us take \(r=\infty \) in (3.4). By (3.8) and (3.9), we obtain
Thus,
From (3.10) and (3.7), we get (3.2) with \(\gamma =\frac{1}{2}-\frac{N}{2q}\).
Finally, it remains to show that \(\mathcal {T}\) maps \(\overline{B}(0,R)\) into \(\overline{B}(0,R)\), with
Indeed, let us take \(v=0\) in (3.2). Then,
with
Now, for every \(\Vert u\Vert _{\mathcal {C}([0,T]; X)}<R\), let \(T\in (0,1)\) be small enough such that \(C_6(R, \varepsilon ) T^{\gamma } <\frac{1}{2} \). Therefore, (3.11) implies
On the other hand, it is not difficult to show that
And
A combination of (3.13) and (3.14) implies
This inequality and (3.12) implies that \(\mathcal {T}\) maps \(\overline{B}(0,R)\) into \(\overline{B}(0,R)\). Thus, we obtain Lemma 6. \(\quad \square \)
Now, by applying Lemma 6, there is a unique mild solution \(u_{\varepsilon , \nu , \kappa }\in \mathcal {C}([0,T]; X)\) (denoted as u for short) satisfying the equation \(\mathcal {T}(u)=u\). This yields Theorem 4.
\(\square \)
Remark 3
By the standard regularity, if \(u_0\) and f are smooth then so is u. Thanks to this point, in what follows, we can use a smoothing effect to the data by assuming that \(u_0\in \mathcal {C}^\infty _c( \mathbb {R}^N)\) and \(f\in \mathcal {C}^\infty _c(Q_T)\).
Next, we derive some estimates for solution u of (3.1) . The first estimate is the \(L^q\)-estimate.
Proposition 2
Let u be a solution of (3.1) in \(Q_T\). Then, for every \(q\in [1, \infty )\) we have
In particular, if \(q=\infty \) then
Moreover, there is a positive constant C, depending only \(T, u_0, f\) such that
Proof
For every \(q > 1\) and for \(t\in (0,T)\), we use \(|u|^{q-2}u\) as a test function to (3.1) and integrate on \(\mathbb {R}^N\) in order to obtain
Thanks to Lemma 3, we get
with \(\psi (u)=|u|^{q-2}u\), and \(\phi (u)=J_{\kappa }(u)\).
On the other hand, we observe that
with
Note that the inequality in (3.20) is also obtained by applying Lemma 3.
A combination of (3.18), (3.19) and (3.20) implies
Using Hölder’s inequality yields
This leads to
with \(y(t)= \displaystyle \int |u(t)|^q \hbox {d}x\). By solving the above OD inequality, we obtain
Again, applying Hölder’s inequality yields (3.15).
Passing to the limit as \(q\rightarrow \infty \), we deduce (3.16).
Next, we prove \(L^1\)-estimate for u.
For any \(\eta >0\), let us put
By multiplying (3.1) with \(\chi _\eta (u)\), and integrating on \(\mathbb {R}^N\), we get
Since \(\chi ^\prime _\eta (u) \geqq 0\), it is clear that
and
by using Lemma 3.
Thus, it follows from (3.21) after integrating on (0, t) that
with
Note that \(\chi _A\) is the characteristic function of the set A.
It is not difficult to verify that
Thus, (3.15) follows with \(q=1\).
It remains to prove (3.17). Using the same argument as above we obtain from (3.18) with \(q=2\) that
Applying Hölder’s inequality yields
Moreover, we have from (3.15) with \(q=2\)
A combination of (3.22) and (3.23) implies (3.17). Then, we obtain Proposition 2.
\(\square \)
Proposition 3
Let u as in Proposition 2. Then, there is a constant \(C=C(m_0, u_0, f)>0\) such that for every \(\kappa , \varepsilon >0\)
Proof
Let us denote \(u=u_\varepsilon \) for short. Now, by using \(J_\kappa (u)\) as a test function to equation (3.1) and integrating both sides on \(Q_T\), we obtain
with
By a simple calculation, we have
Note that \(J^\prime (s)\geqq 0\), so
and
with
We observe that \(W^\prime (s) \geqq 0\), so inequality (3.28) is obtained by using the Stroock–Varopoulos’s inequality as (3.20).
Thus, it follows from (3.25), (3.26), (3.27) and (3.28) that
Furthermore, thanks to Proposition 2, we have
Combining (3.29) and (3.30) yields (3.24). This completes the proof of Proposition 3. \(\quad \square \)
Remark 4
As a consequence of (3.16), the norm \(\Vert u(t)\Vert _{L^\infty ( \mathbb {R}^N)}\) cannot be explosive for \(t<T\). Furthermore, we get the global existence of solution u provided that \(f\in L^\infty (Q_\infty ) \cap L^1(Q_\infty )\). In particular, if \(f\equiv 0\) the norm \(\Vert u(t)\Vert _{L^q( \mathbb {R}^N)}\) is nonincreasing with respect to t for any \(q\geqq 1\).
Remark 5
We emphasize that for any given \(\delta _1, \delta _2>0\) the right hand side of the estimates in Propositions 2 and 3 are independent of \(\varepsilon , \nu , \kappa \). Moreover, the two perturbation terms \(-\delta _1 \Delta u\) and \(\delta _2 (-\Delta )^{s_0}[|u|^{m_0-1} u]\) are positive and play a role in absorbing \( \text{ div } \left( |u|^{m_1} \nabla (-\Delta )^{-s}[|u|^{m_2-1}u]\right) \). This observation will enable us to pass to the limit as \(\varepsilon , \nu , \kappa \rightarrow 0\) in what follows.
3.1 Limit as \(\varepsilon \rightarrow 0\)
Next, we shall pass to the limit as \(\varepsilon \rightarrow 0\).
Proposition 4
Let \(u_{\varepsilon }\) be the solution of problem (3.1), obtained from Lemma 6. Then, there exists a subsequence of \(\{u_{\varepsilon }\}_{\varepsilon >0}\) (still denoted as \( \{u_{\varepsilon }\}_{\varepsilon >0}\) ) such that, for any \(R>0\),
Furthermore, \(u\in L^1(Q_T)\cap L^\infty (Q_T)\cap L^2(0,T; H^1(\mathbb {R} ^N))\) is a solution of the following problem:
In addition, there exists a positive constant \(C=C(u_0, f_0, m_0)\) such that
Proof
The main idea of the proof is to pass to the limit as \(\varepsilon \rightarrow 0\) in the equation satisfied by \(u_\varepsilon \)
for all \(\varphi \in \mathcal {C}^\infty _c (Q_T)\). Here, we denote
At the beginning, let us fix a test function \(\varphi \in \mathcal {C}^\infty _c (Q_T)\) such that \(Supp(\varphi )\subset B_R\), for \(R>0\). Now, we recall a compactness result of Simon [25], used several times in what follows.
Lemma 7
Assume that the spaces \(V_1\subset V_2\subset V_3\) with compact embedding \(V_1 \subset V_2\) . Let \(\{u_n\}_{n\geqq 1}\) be a bounded sequence in \(L^p(0,T; V_1)\) and let \(\{ \partial _t u_n \}_{n\geqq 1}\) be bounded in \(L^1(0,T; V_3)\). Then \(\{u_n\}_{n\geqq 1}\) is relatively compact in \(L^p(0,T; V_2)\).
Next, we have the following uniform estimates:
Lemma 8
\({\text {div}}\left( \Theta _{\varepsilon ,\nu }(u_\varepsilon )\right) \) and \(\mathcal {L}^{s_0}_\varepsilon [J_\kappa (u_\varepsilon )] \) are uniformly bounded in \(L^{2}(0,T; H^{-1}(B_R))\) with respect to \(\varepsilon , \kappa >0\), where \(H^{-1}(B_R)\) is the dual space of \(H^1_0(B_R)\).
Proof of Lemma 8
In fact, we have
The equality in (3.34) is obtained by using the Plancherel theorem. By Hölder’s inequality, we get
Moreover, applying Hölder’s inequality and Young’s inequality yields
A combination of (3.35) and (3.36) gives that
It follows from the last inequality and (3.24) that \(\mathcal {L}^{s_0}_\varepsilon [J_\kappa (u_\varepsilon )] \) is bounded in \(L^2(0,T; H^{-1}(B_R))\) by a constant, not depending on \(\varepsilon , \kappa \).
Next, we claim that \(\Vert {\text {div}} \Theta _{\varepsilon ,\nu }(u_\varepsilon ) \Vert _{L^2\left( 0,T; H^{-1}(B_R) \right) } \) is bounded by a constant, being independent of \(\varepsilon \). Due to some technical reasons, we divide our proof into the two following cases:
(i) If \(\frac{1}{2}\leqq s<1\), for any \(t>0\) we apply the Plancherel theorem and Hölder’s inequality in order to obtain
where \(q_s=\frac{2N}{N-2(2s-1)}\), and \(q_s'=\frac{q_s}{q_s-1}\). Note that \(q_s\geqq 2\).
According to Proposition 2, \(\Vert u_\varepsilon \Vert ^{m_1}_{L^\infty (Q_T)}\) is bounded by a constant, not depending on \(\varepsilon \). Thus, it suffices to prove that \( \Vert \nabla (-\Delta )^{-1} \mathcal {L}_\varepsilon ^{1-s} [G_\nu (u_\varepsilon (t))] \Vert _{L^{q_s}(\mathbb {R}^N)}\) is uniformly bounded for all \(\varepsilon >0\), and for \(t>0\). Indeed, it follows from the Riesz potential estimate that
Morever, it is clear that
It follows from Proposition 2, and the Interpolation theorem that \(\Vert u^{m_2+2}_\varepsilon (t) \Vert _{L^2(\mathbb {R}^N)}\) is uniformly bounded for all \(t>0\), and all \(\varepsilon >0\).
From (3.37), (3.38), and the last inequality, we get the claim for the case \(s\in [\frac{1}{2}, 1) \).
In the following, we remove the dependence on time t of the terms in our estimates for brief if no confusion.
(ii) If \(0<s<\frac{1}{2}\), we then have from the Plancherel theorem and Hölder’s inequality that
On the other hand, using the Plancherel theorem yields
Since \(0<s<\frac{1}{2}\), we can apply Hölder’s inequality in order to obtain
Combining (3.40) and (3.41) yields
Since \(G_\nu \) is a Lipschitz function with \(G_\nu (0)=0\), and by (3.17), there exists a constant \(C>0\), being independent of \(\varepsilon \) such that
Thus, the claim follows from (3.42) and (3.43).
This puts an end to the proof of Lemma 8. \(\quad \square \)
Now, thanks to Lemma 8, \(\partial _t u_\varepsilon \) is bounded in \(L^2(0,T; H^{-1}(B_R))\) by a constant not depending on \(\varepsilon \). Moreover, it follows from Proposition 2 that \(u_\varepsilon \) is bounded in \(L^2(0,T; H^1_0(B_R))\). Thus, applying Lemma 7 implies that there is a subsequence of \(\{u_\varepsilon \}_{\varepsilon >0}\) (still denoted as \(\{u_\varepsilon \}_{\varepsilon >0}\)) such that
Thanks to Proposition 2, we deduce
and
By (3.17), \(\nabla u_\varepsilon \) converges weakly to \(\nabla u\) in \(L^2(B_R\times (0,T))\) up to a subsequence. Thus, we get
Next, we consider the difference between the two integrals as follows:
We claim that
as \(\varepsilon \rightarrow 0\), with
In fact, it is obvious that \( A_\varepsilon (\xi ) \rightarrow 0\) almost everywhere in \(Q_T\).
Moreover, we have
The last inequality is obtain by using the fact \(| \mathcal {F}\{\mathcal {L}^{s}_\varepsilon \} (\xi )| \leqq |\mathcal {F} \{ (-\Delta )^{s} (\xi )| = C|\xi |^{2s}\), for every \(\xi \in \mathbb {R}^N\), and for \(s\in (0,1)\).
Furthermore, using the standard property of Fourier transform yields
by (3.15). Similarly, we obtain
Thus,
Since \(\varphi (t)\in \mathcal {S}(\mathbb {R}^N)\), so is \(\mathcal {F}\{\varphi (t)\}\). This fact implies that \( |\xi |^{2 s_0} |\mathcal {F}\{\varphi (t)\}(\xi )| \) is integrable on \(Q_T\). Thanks to the Dominated Convergence Theorem, we obtain (3.47).
This leads to
It remains to prove that
as \(\varepsilon \rightarrow 0\). For technical reasons, we divide our proof into the two following cases:
-
If \(\frac{1}{2}\leqq s<1\), we rewrite
$$\begin{aligned}&\int ^T_0 \int \left( {\text {div}} \Theta _{\varepsilon ,\nu } (u_\varepsilon ) - {\text {div}} \left( H_\nu (u) \nabla (-\Delta )^{-s} [G_\nu (u)]\right) \right) \varphi \hbox {d}x\hbox {d}t \\&\quad = \int ^T_0 \int \left( H_\nu (u_\varepsilon ) \nabla (-\Delta )^{-1} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] \right. \\&\qquad - \left. H_\nu (u) \nabla (-\Delta )^{-s} [G_\nu (u)] \right) .\nabla \varphi \hbox {d}x\hbox {d}t \end{aligned}$$Put
$$\begin{aligned} A_1 = \int ^T_0 \int \left| H_\nu (u_\varepsilon ) -H_\nu (u) \right| \left| \nabla (-\Delta )^{-1} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] \right| | \nabla \varphi | \hbox {d}x\hbox {d}t , \end{aligned}$$and
$$\begin{aligned} A_2 = \int ^T_0 \int |H_\nu (u)| \left| \nabla (-\Delta )^{-1} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] - \nabla (-\Delta )^{-s} [G_\nu (u)]\right| |\nabla \varphi | \hbox {d}x\hbox {d}t. \end{aligned}$$To obtain (3.49), it suffices to show that \(A_1, A_2\rightarrow 0 \), as \(\varepsilon \rightarrow 0\).
$$\begin{aligned} A_1&= \int ^T_0 \int \left| H_\nu (u_\varepsilon ) -H_\nu (u) \right| \left| \nabla ^{1-2s} (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] \right| | \nabla \varphi | \hbox {d}x\hbox {d}t \\&= \int ^T_0 \int \left| H_\nu (u_\varepsilon ) -H_\nu (u) \right| \left| \mathcal {I}_{2s-1}\left[ (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )]\right] \right| | \nabla \varphi | \hbox {d}x\hbox {d}t . \end{aligned}$$By the fundamental estimate for the Riesz potential and the Plancherel theorem, we get
$$\begin{aligned} \Vert \mathcal {I}_{2s-1} \left[ (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )]\right] \Vert _{L^{q_s}}&\lesssim \Vert (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )]\Vert _{L^2} \\&\lesssim \Vert G_\nu (u_\varepsilon ) \Vert _{L^2}, \end{aligned}$$with \(q_s=\frac{2N}{N-2(2s-1)} \geqq 2\).
Again, we observe that \(\Vert G_\nu (u_\varepsilon ) \Vert _{L^2}\) is bounded by a constant not depending on \(\varepsilon \). This implies that the term \(\mathcal {I}_{2s-1} \left[ (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )]\right] \) is also bounded in \(L^{q_s}\). Moreover, it is not difficult to prove that \(H_\nu (u_\varepsilon )\rightarrow H_\nu (u)\) in \(L^p\left( B_R\times (0,T)\right) \), for any \(p\in [ 1, \infty )\). Thus, \(A_1 \rightarrow 0\) as \(\varepsilon \rightarrow 0\).
Similarly, we have
$$\begin{aligned}&\Vert \mathcal {I}_{2s-1} \left[ (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] - G_\nu (u) \right] \Vert _{L^{q_s}} \\&\quad \lesssim \Vert (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] - G_\nu (u) \Vert _{L^2}. \end{aligned}$$Applying Lemma 2 yields
$$\begin{aligned} \Vert (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] - G_\nu (u) \Vert _{L^2}\rightarrow 0 \end{aligned}$$as \(\varepsilon \rightarrow 0\). Thus
$$\begin{aligned} \Vert \mathcal {I}_{2s-1} \left[ (-\Delta )^{-(1-s)} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] - G_\nu (u) \right] \Vert _{L^{q_s}}\rightarrow 0 . \end{aligned}$$This implies \(A_2\rightarrow 0\).
-
If \(s\in (0,\frac{1}{2})\), we write
$$\begin{aligned}&\int ^T_0 \int {\text {div}} (\Theta _{\varepsilon ,\nu }(u_\varepsilon ) ) \varphi \hbox {d}x\hbox {d}t \nonumber \\&\quad = \int ^T_0 \int {\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) (-\Delta )^{-1} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] \hbox {d}x\hbox {d}t. \end{aligned}$$(3.50)We will show that
$$\begin{aligned}&\int ^T_0 \int {\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) (-\Delta )^{-1} \mathcal {L}^{1-s}_{\varepsilon }[G_\nu (u_\varepsilon )] \hbox {d}x\hbox {d}t \nonumber \\&\quad \rightarrow \int ^T_0 \int {\text {div}} \left( H_\nu (u) \nabla \varphi \right) (-\Delta )^{-s} [G_\nu (u)] \hbox {d}x\hbox {d}t\nonumber \\&\quad = \int ^T_0 \int {\text {div}} \left( H_\nu (u) \nabla (-\Delta )^{-s} [G_\nu (u)] \right) \varphi \hbox {d}x\hbox {d}t . \end{aligned}$$(3.51)On one hand, we see that
$$\begin{aligned} \Vert {\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) \Vert _{L^2} \leqq \Vert H^\prime _\nu (u_\varepsilon ) \nabla u_\varepsilon . \nabla \varphi \Vert _{L^2} + \Vert H_\nu (u_\varepsilon ) \Delta \varphi \Vert _{L^2} . \end{aligned}$$(3.52)It follows from Proposition 2 that \(H^\prime _\nu (u_\varepsilon )\) and \(H_\nu (u_\varepsilon )\) are bounded by a constant, being independent of \(\varepsilon \). This implies that the right hand side of (3.52) is also bounded, and so is \(\Vert {\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) \Vert _{L^2}\). On the other hand, it is not difficult to verify that
$$\begin{aligned} {\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) \rightarrow {\text {div}} \left( H_\nu (u) \nabla \varphi \right) , \quad \text {in } ~\mathcal {D}'(Q_T) .\end{aligned}$$Then, \({\text {div}} \left( H_\nu (u_\varepsilon ) \nabla \varphi \right) \) converges weakly to \({\text {div}} \left( H_\nu (u) \nabla \varphi \right) \) in \(L^2(B_{R})\) (up to a subsequence).
Therefore, it is sufficient to prove that
Indeed, we have
with \(q^{\star }_s= \frac{2N}{N-4s}>2\).
Since \(G_\nu (u_\varepsilon )\rightarrow G_\nu (u)\) strongly in \(L^2(Q_T)\), then a modification of Lemma 2 implies
By applying Hölder’s inequality, and by (3.55), we obtain
This implies (3.53).
As a consequence, we obtain (3.51) and (3.49) alternatively.
A combination of (3.45), (3.48) and (3.49) ensures that u is a weak solution of (3.1).
Next, we prove (3.32). Indeed, we can mimic the proof of (3.47) to obtain
as \(\varepsilon \rightarrow 0\). Furthermore, it follows from (3.24) that
Then,
which implies (3.32).
To complete the proof of Proposition 4, it remains to show that \(u\in \mathcal {C}([0,T]; L^p(\mathbb {R}^N))\), for all \(p\geqq 1\). By (3.17), we observe that \(u_\varepsilon \) is bounded in \(L^2(0,T; H^1_0(B_R))\) by a constant, not depending on \(\varepsilon \). Moreover, \(\partial _t u_\varepsilon \) is also bounded in \(L^1(Q_T) + L^2(0,T; H^{-1}(B_R))\), for any \(R>0\). Thanks to Theorem 1.1, [22], we obtain
From this fact, we can mimic the argument given in page 21 of [15] in order to get
By the boundedness of \(u_\varepsilon \) in \(Q_T\), we have \(u\in \mathcal {C}\left( [0,T]; L^p(\mathbb {R}^N)\right) \), for every \(p\in [1,\infty )\).
This ends to the proof of Proposition 4. \(\quad \square \)
Remark 6
It is not difficult to verify that the solution u, obtained by passing to the limit as \(\varepsilon \rightarrow 0\) also satisfies Proposition 2. Moreover, the estimates in this part are independent of \(\kappa \). This observation will allow us to pass to the limit as \(\kappa \rightarrow 0\) in what follows.
3.2 Limit as \(\kappa \rightarrow 0\)
In this part, we shall pass to the limit as \(\kappa \rightarrow 0 \).
Proposition 5
Let \(u_{\kappa }\) be the solution of problem (3.31), obtained in Proposition 2. Then, it holds that for any \(R>0\),
up to a subsequence.
Furthermore, \(u\in L^1(Q_T)\cap L^\infty (Q_T)\cap L^2(0,T; H^1(\mathbb {R} ^N))\) is a solution of the following problem:
where we use the notation \(\Theta _\nu (u)= H_\nu (u) \nabla (-\Delta )^{-s} [G_\nu (u)]\). In addition, we have
where \(C>0\) depends only on \(m_0, u_0, f\).
Proof
We first note that \({\text {div}} \Theta _\nu (u_\kappa )\) and \((-\Delta )^{s_0}[J_\kappa (u_\kappa )]\) are bounded in \(L^2(0,T; H^{-1}(B_R))\) by a constant not depending on \(\kappa \), see Remark 6. Thanks to the compactness result in Lemma 7, there is a subsequence of \(\{u_\kappa \}_{\kappa >0}\) such that
as \(\kappa \rightarrow 0\). It follows from Proposition 2 that
and
Now, it suffices to show that u satisfies equation (3.56) in the weak sense.
Indeed, it is not difficult to verify that
Thus, it remains to demonstrate that
Indeed, we have, from that Plancherel’s theorem, that
By (3.16), we have
Moreover, it is clear that \(J_\kappa (u_\kappa ) \rightarrow |u|^{m_0-1} u\) as \(\kappa \rightarrow 0\), for almost everywhere \((x,t)\in Q_T\).
Thus, applying the Dominated Convergence Theorem yields
when \(\kappa \rightarrow 0\). This implies (3.59).
In conclusion, u is a weak solution of problem (3.56).
Finally, (3.57) follows from (3.32), and we obtain the proof of Proposition 5.
\(\square \)
3.3 Limit as \(\nu \rightarrow 0\)
Proposition 6
Let \(u_{\nu }\) be the solution, obtained in Proposition 5. Then, there exists a subsequence of \( \{u_{\nu }\}_{\nu >0}\) converging to a function u in \(L^2\left( B_R\times (0,T) \right) \) for any \(R>0\). Moreover, \(u\in L^1(Q_T)\cap L^\infty (Q_T) \cap L^2( 0,T ; H^1(\mathbb {R} ^N))\) is a solution of the equation
Recall here that \(\Theta (u)= H(u) \nabla (-\Delta )^{-s}[G(u)]\), with \( H(u)=|u|^{m_1}\) and \(G(u)= |u|^{m_2-1} u\).
Proof
By (3.57) and the same argument as in Lemma 8, we obtain \(\delta _2 (-\Delta )^{s_0} (|u_\nu |^{m_0-1} u_\nu )\) is bounded in \(L^2(0,T; H^{-1}(B_R))\) by a constant not depending on \(\nu \).
Now, we show that
with \(C>0\) is independent of \(\nu \).
The idea of the proof of (3.61) is most likely to the one of Lemma 8, but we need to derive the estimates, not depending on the parameter \(\nu \). To do that, we have to use some properties of the term \((-\Delta )^{s_0}(|u_\nu |^{m_0-1}u_\nu )\). We divide our proof into the two following cases:
-
(i) If \(s\in [\frac{1}{2}, 1)\), we mimic the proof of (3.37) and (3.38) to obtain
$$\begin{aligned}&\Vert {\text {div}} \left( H_\nu (u_\nu )\nabla (-\Delta )^{-s} {[}G_\nu (u_\nu )] \right) \Vert _{H^{-1} (B_R)}\nonumber \\&\quad \leqq \Vert u_\nu \Vert ^{m_1}_{L^\infty (Q_T)} \Vert \mathcal {I}_{2s-1} {[}G_\nu (u_\nu )] \Vert _{L^2(B_R)}, \end{aligned}$$(3.62)and
$$\begin{aligned} \Vert \mathcal {I}_{2s-1} [G_\nu (u_\nu )] \Vert _{L^{q_s}}\lesssim \Vert G_\nu (u_\nu ) \Vert _{L^2}\leqq \left( \int |u_\nu (x)|^{2m_2} \hbox {d}x\right) ^\frac{1}{2}. \end{aligned}$$(3.63)From the Sobolev embedding, we have
$$\begin{aligned} \Vert |u_\nu |^{m_0-1} u_\nu \Vert _{L^{q^\star _{s_0}}}\lesssim \Vert (-\Delta )^{\frac{s_0}{2}}( |u_\nu |^{m_0-1} u_\nu ) \Vert _{L^2}, \end{aligned}$$with \(q^\star _{s_0}=\frac{2N}{N-2 s_0}\). It follows from (3.57) that there exists a constant \(C>0\) (not depending on \(\nu \)) such that
$$\begin{aligned} \int |u_\nu (x)|^{m_0 q^\star _{s_0} } \hbox {d}x\leqq C. \end{aligned}$$(3.64)Since \(m_0\leqq \frac{m_2(N-2s_0)}{2N}\), then we have from (3.64)
$$\begin{aligned} \int |u_\nu (x)|^{2m_2} \hbox {d}x \leqq \Vert u_\nu \Vert ^{2m_2-m_0 q^\star _{s_0}}_{L^\infty (Q_T)} \int |u_\nu (x)|^{m_0 q^\star _{s_0} } \hbox {d}x \leqq C, \end{aligned}$$(3.65)We also remind here that \(\Vert u_\nu \Vert _{L^\infty (Q_T)}\) is bounded by a constant \(C=C(u_0, f)\).
A combination of (3.62), (3.63) and (3.65) gives that
$$\begin{aligned} \Vert {\text {div}} \left( H_\nu (u_\nu )\nabla (-\Delta )^{-s} [G_\nu (u_\nu )] \right) \Vert _{H^{-1}(B_R)} \leqq C. \end{aligned}$$Alternatively, we obtain (3.61).
-
(ii) If \(s\in (0,\frac{1}{2})\) then
$$\begin{aligned}&\Vert {\text {div}} \left( H_\nu (u_\nu )\nabla (-\Delta )^{-s} [G_\nu (u_\nu )] \right) \Vert _{H^{-1}(B_R)} \nonumber \\&\quad = \sup _{ \Vert \psi \Vert _{H^1_0(B_R)} = 1} \left| \int _{B_R} \Theta (u_\nu ) \cdot \nabla \psi \hbox {d}x \right| \nonumber \\&\quad =\sup _{\Vert \psi \Vert _{H^1_0(B_R)}=1 } \left| \int _{\mathbb {R}^N} {\text {div}} (H_\nu (u_\nu ) \nabla \psi ) (-\Delta )^{-s} [G_\nu (u_\nu )] \hbox {d}x \right| \nonumber \\&\quad = \sup _{\Vert \psi \Vert _{H^1_0(B_R)}=1 } \left| \int _{\mathbb {R}^N} (-\Delta )^{-\frac{1}{2}} {\text {div}} ( H_{\nu } (u_{\nu }) \nabla \psi ) (-\Delta )^{\frac{1}{2}-s}[G_{\nu }(u_{\nu })] \hbox {d}x \right| \nonumber \\&\quad \leqq \sup _{\Vert \psi \Vert _{H^1_0(B_R)}=1 } \Vert (-\Delta )^{\frac{1}{2}-s}G_{\nu }(u_{\nu })\Vert _{L^2(\mathbb {R}^N)} \Vert (-\Delta )^{-\frac{1}{2}} {\text {div}}(H_{\nu }(u_{\nu })\nabla \psi ) \Vert _{L^2(\mathbb {R}^N)} . \end{aligned}$$(3.66)Thanks to Plancherel’s theorem, there is a constant \(C=C(N)>0\) such that
$$\begin{aligned} \Vert (-\Delta )^{-\frac{1}{2}} {\text {div}}(H_{\nu }(u_{\nu })\nabla \psi ) \Vert _{L^2(\mathbb {R}^N)} \leqq C \Vert H_{\nu }(u_{\nu })\nabla \psi \Vert _{L^2(\mathbb {R}^N)} \leqq C \Vert \psi \Vert _{H^1_0(B_R)}. \end{aligned}$$(3.67)By (3.66) and (3.67), we obtain
$$\begin{aligned}&\int ^T_0 \Vert {\text {div}} \left( H_\nu (u)\nabla (-\Delta )^{-s} [G_\nu (u)] \right) \Vert ^2_{H^{-1}(B_R)} \hbox {d}t \nonumber \\&\quad \leqq C \int ^T_0 \Vert (-\Delta )^{\frac{1}{2}-s}G_{\nu }(u_{\nu })\Vert ^2_{L^2(\mathbb {R}^N)} \hbox {d}t. \end{aligned}$$(3.68)On the other hand, we have
$$\begin{aligned} |G_{\nu }(a)-G_{\nu }(b)| \leqq C \left| |a|^{m_0-1}a- |b|^{m_0-1} b \right| (|a|+|b|)^{m_2-m_0}, \quad \forall a, b\in \mathbb {R}, \end{aligned}$$with \(C=C(m_0, m_2)\), for all \(\nu \in (0,1)\). Thus,
$$\begin{aligned}&\Vert (-\Delta )^{\frac{1}{2}-s}G_{\nu } (u_\nu ) \Vert _{L^2(Q_T)}^2 = \int _{0}^{T} \int \int \frac{|G_{\nu } (u_{\nu }(x)) -G_{\nu } (u_{\nu }(y)) |^2}{|x-y|^{N+2(1-2s)}} \hbox {d}x \hbox {d}y \hbox {d}t \nonumber \\&\quad \leqq C \Vert u_\nu \Vert ^{m_2-m_0}_{L^\infty (Q_T)} \int _{0}^{T}\int \int \frac{\left| |u_{\nu }(x)|^{m_0-1}u_{\nu }(x) -|u_{\nu }(y)|^{m_0-1}u_{\nu }(y) \right| ^2}{ |x-y|^{N+2(1-2s)} } \hbox {d}x\hbox {d}y\hbox {d}t \nonumber \\&\quad \leqq C \Vert u_\nu \Vert ^{m_2-m_0}_{L^\infty (Q_T)}\Vert (-\Delta )^{\frac{s_0}{2}} \left( |u_{\nu }|^{m_0-1}u_{\nu } \right) \Vert _{L^2(Q_T)}^2 . \end{aligned}$$(3.69)Note that \(s_0=1-2s\) in this case. Thanks to (3.57) and Proposition 2, the right hand side of the last inequality is bounded by a constant, that is independent of \(\nu \), and so is \(\Vert (-\Delta )^{\frac{1}{2}-s}G_{\nu } (u_\nu ) \Vert _{L^2(Q_T)}^2\).
Thus, (3.61) follows from (3.68) and the boundedness of \(\Vert (-\Delta )^{\frac{1}{2}-s}G_{\nu } (u_\nu ) \Vert _{L^2(Q_T)}^2\). Thanks to Lemma 7, there is a subsequence of \(\{u_\nu \}_{\nu >0}\), converging to u in \(L^2(B_R\times (0,T))\) when \(\nu \rightarrow 0\).
By Proposition 2, we deduce
and
Now, we shall show that u is a weak solution of problem (3.60).
We claim that
strongly in \(L^2(Q_T)\).
It follows from Plancherel’s theorem that
Thus,
Since \(H_\nu (u_\nu (x,t))\rightarrow | u(x,t)|^{m_1}\) for almost everywhere \((x,t)\in Q_T\) (up to a subsequence if necessary), then we have
Furthermore, by Proposition 2 we have
Thanks to the Dominated Convergence Theorem, we obtain (3.70).
Next, we deduce from (3.69) that
weakly in \(L^2(Q_T)\) as \(\nu \rightarrow 0\).
Thus,
On the other hand, it is not difficult to show that
as \(\nu \rightarrow 0\). Therefore, u is a weak solution of problem (3.60).
Or, we complete the proof of Proposition 6. \(\quad \square \)
3.4 Limit as \(\delta _1, \delta _2 \rightarrow 0\)
In this subsection, we will pass to the limit as \(\delta _2, \delta _1 \rightarrow 0\) alternatively. Then, we have the following result:
Proposition 7
Let \(u_{\delta _2}\) be a solution of (3.60) above. Then, there exists a subsequence of \(\{u_{\delta _2}\}_{\delta _2>0}\), converging to a function u in \(L^2\left( B_R\times (0,T)\right) \) for any \(R>0 \). Moreover, \(u\in L^1(Q_T)\cap L^\infty (Q_T)\) is a weak solution of the following problem:
Proof
We rewrite equation (3.60), satisfied by \(u_{\delta _2}\) in the weak sense as follows:
Our purpose is to pass to the limit as \(\delta _2\rightarrow 0\) in (3.72) in order to obtain
which says that u is a weak solution of equation (3.71).
First, we claim that
as \(\delta _2\rightarrow 0\).
Indeed, for any \(\varphi \in \mathcal {D}(\mathbb {R}^N\times (0,T))\), we apply Hölder’s inequality and (3.57) in order to obtain
This yields the claim.
Next, we prove that there is a subsequence of \(\{u_{\delta _2}\}_{\delta _2 >0}\), (still denoted as \(\{u_{\delta _2}\}_{\delta _2 >0}\)) such that
for any \(R>0\), and for any \(q\in [1,\infty )\). Thus, up to a subsequence, we have
so
In fact, using \(|u|^{q-2}u\) as a test function to equation (3.60), we obtain as in (3.18)
Let us fix \(q>1\) in (3.77) such that \(\gamma =\frac{ m_1+ m_2 + q-1}{2} \geqq 1\) . It follows from Lemma 5 and (3.77) that
where \(C>0\) is independent of \( \delta _2, \delta _1\).
This implies that \(v_{\delta _2}=|u_{\delta _2}|^{\gamma -1} u_{\delta _2}\) is uniformly bounded in \(L^2(0,T; H^{1-s}(\mathbb {R}^N))\) for all \(\delta _2>0\).
Moreover, since \(\gamma \geqq 1\), it follows then from Proposition 2 and the Interpolation theorem that
and \(\Vert v_{\delta _2}\Vert _{L^2\left( \mathbb {R}^N\times (0,T)\right) }\) is bounded by a constant, being independent of \(\delta _1, \delta _2\). Thus, there is a subsequence of \(\{v_{\delta _2}\}_{\delta _2>0}\) (still denoted as \(\{v_{\delta _2}\}_{\delta _2>0}\)) such that
Thanks to a result of Rakotoson and Temam [23], we obtain, for any \(R>0\), that
Now, applying Lemma 4 and Hölder’s inequality yields
A combination of the last inequality, (3.79), and the uniform boundedness of \(u_{\delta _2}\) implies (3.75).
It remains to show the convergence of \(\nabla (-\Delta )^{-s}[G(u_{\delta _2})] \rightarrow \nabla (-\Delta )^{-s}[G(u)] \) in \(\mathcal {D}'(Q_T)\). We divide our proof into the two following cases:
-
(i) The case \(s\in (0 ,\frac{1}{2})\). We show that
$$\begin{aligned} \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \rightharpoonup \nabla (-\Delta )^{-s}[G(u)],\quad \text {in } L^p\left( 0,T; W^{-1, p}(\mathbb {R}^N) \right) \nonumber \\ \end{aligned}$$(3.80)up to a subsequence, for \(p>1\) such that \( m_2 p' \geqq 1\), \(\frac{1}{p}+ \frac{1}{p'}=1\), and \(W^{-1, p}(\mathbb {R}^N)\) is the dual space of \(W^{1, p}(\mathbb {R}^N)\).
We emphasize that it is enough to consider the case \(0<m_2<1\) in the following because the case \(m_2\geqq 1\) is much easier.
By using Hölder’s inequality and Plancherel’s theorem, we get
$$\begin{aligned}&\Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{W^{-1,p}(\mathbb {R}^N)} \\&\quad = \sup _{\Vert \psi \Vert _{W^{1, p}(\mathbb {R}^N)} = 1} \left| \int _{\mathbb {R}^N} \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \psi \hbox {d}x \right| \\&\quad = \sup _{\Vert \psi \Vert _{ W^{1,p} ( \mathbb {R}^N ) }=1} \left| \int _{\mathbb {R}^N} G(u_{\delta _2}) \nabla (-\Delta )^{-s}\psi \hbox {d}x \right| \\&\quad \leqq \sup _{\Vert \psi \Vert _{ W^{1,p} ( \mathbb {R}^N ) }=1} \Vert G(u_{\delta _2}) \Vert _{L^{p'}(\mathbb {R}^N)} \Vert \nabla ^{1-2s} \psi \Vert _{L^p(\mathbb {R}^N)} . \end{aligned}$$Now, we apply Lemma 11 in order to obtain
$$\begin{aligned} \Vert \nabla ^{1-2s} \psi \Vert _{L^p(\mathbb {R}^N)} \leqq C \Vert \nabla \psi \Vert ^{1-2s}_{L^p(\mathbb {R}^N)} \Vert \psi \Vert ^{2s}_{L^p(\mathbb {R}^N)} . \end{aligned}$$Combining the two last inequalities yields
$$\begin{aligned} \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{W^{-1,p}(\mathbb {R}^N)}\lesssim \Vert G(u_{\delta _2}) \Vert _{L^{p'}(\mathbb {R}^N)} =\Vert u_{\delta _2} \Vert ^{m_2}_{L^{m_2 p'}(\mathbb {R}^N)} .\nonumber \\ \end{aligned}$$(3.81)Thanks to Proposition 2 and the fact \(m_2 p'\geqq 1\), it follows from (3.81) that
$$\begin{aligned} \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{L^p \left( 0,T; W^{-1,p}(\mathbb {R}^N) \right) } \leqq C(u_0, f, p, m_2, T). \end{aligned}$$(3.82)Thus, \( \nabla (-\Delta )^{-s}[G(u_{\delta _2})]\) is uniformly bounded in \(L^p \left( 0,T; W^{-1,p}(\mathbb {R}^N) \right) \) for all \(\delta _2>0\). Then, there exists a function w such that
$$\begin{aligned} \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \rightharpoonup w,\,\, \text {in } L^p \left( 0,T; W^{-1,p}(\mathbb {R}^N) \right) , \end{aligned}$$up to a subsequence.
On the other hand, we also have
$$\begin{aligned} G(u_{\delta _2}) \rightharpoonup G(u),\,\, \text {in }\, L^{p'}\left( Q_T\right) . \end{aligned}$$This implies (3.80).
-
(ii) The case \(s\in (\frac{1}{2}, 1)\). We prove that for any \(R>0\)
$$\begin{aligned} \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \rightharpoonup \nabla (-\Delta )^{-s}[G(u)], \,\, \text {in }\, L^2\left( B_R\times (0,T)\right) . \end{aligned}$$(3.83)To do this, we first show that
$$\begin{aligned} \int ^T_0 \Vert \nabla (-\Delta )^{-s} [G(u_{\delta _2})] \Vert ^2_{ L^2(B_R) } \hbox {d}t \leqq C, \end{aligned}$$(3.84)where \(C>0\) is independent of \(\delta _1, \delta _2\).
Let us fix \(q>1\) in (3.77) such that \(\gamma \geqq 1\) in (3.78). For any \(\beta \in (s,\frac{1+s}{2})\), we apply Hölder’s inequality and the Plancherel theorem to get
$$\begin{aligned}&\Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{L^{r}(\mathbb {R}^N)} \\&\quad = \sup _{\Vert \psi \Vert _{ L^{r'}(\mathbb {R}^N)}=1} \left| \int _{\mathbb {R}^N} (-\Delta )^{\beta -s}[G(u_{\delta _2})] \nabla (-\Delta )^{-\beta } \psi \hbox {d}x \right| \\&\quad \leqq \sup _{\Vert \psi \Vert _{ L^{r'} ( \mathbb {R}^N ) }=1} \Vert (-\Delta )^{\beta -s} [ G(u_{\delta _2}) ] \Vert _{L^{p'}(\mathbb {R}^N)} \Vert \mathcal {I}_{2\beta -1} (\psi ) \Vert _{L^{p}(\mathbb {R}^N)}\\&\quad \lesssim \sup _{\Vert \psi \Vert _{ L^{r'}( \mathbb {R}^N ) }=1} \Vert (-\Delta )^{\beta -s} [ G(u_{\delta _2}) ] \Vert _{L^{p'}(\mathbb {R}^N)} \Vert \psi \Vert _{L^\frac{Np}{N+p(2\beta -1)}(\mathbb {R}^N)} , \end{aligned}$$provided that
$$\begin{aligned} \frac{Np}{N+p(2\beta -1)}>1 \Leftrightarrow \frac{N}{p'} > (2\beta -1). \end{aligned}$$(3.85)Now, we take \(r'=\frac{Np}{N+p(2\beta -1)}\) in the last inequality in order to get
$$\begin{aligned} \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{L^{r}(\mathbb {R}^N)} \lesssim \Vert (-\Delta )^{\beta -s} [ G(u_{\delta _2}) ] \Vert _{L^{p'}(\mathbb {R}^N)} . \end{aligned}$$(3.86)Next, let us put \(\gamma _0=\frac{m_2}{\gamma }\in (0,1)\), and let \(\beta \) be such that \(\beta -s < \gamma _0 (1-s)\).
Applying Lemma 12 with \(v= |u_{\delta _2}|^\gamma sign (u_{\delta _2})\), and \(\Gamma (v)= |v|^{\gamma _0} sign(v)\) yields
$$\begin{aligned} \Vert (-\Delta )^{\beta -s} [ G(u_{\delta _2}) ] \Vert _{L^{p'}(\mathbb {R}^N)}\leqq C \Vert |u_{\delta _2}|^{\gamma -1} u_{\delta _2} \Vert _{\dot{H}^{1-s} (\mathbb {R}^N) } , \end{aligned}$$with
$$\begin{aligned} \frac{\beta -s}{\gamma _0} +\frac{N}{2}= \frac{N}{p'}+ 1-s . \end{aligned}$$(3.87)Since \(N\geqq 2\), \(\gamma _0\in (0,1)\), and \(\beta \in (s,\frac{1+s}{2})\), then it is not difficult to verify that there exists a real number \(p'\in (2,\frac{N}{2\beta -1})\) so that (3.85) and (3.87) hold.
Combining the last inequalities and (3.86) yields
$$\begin{aligned} \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert _{L^{r}(\mathbb {R}^N)} \leqq C \Vert |u_{\delta _2} |^{\gamma -1} u_{\delta _2} \Vert _{\dot{H}^{1-s} (\mathbb {R}^N)} . \end{aligned}$$Here, we note that \(r>2\) since \(\beta \) is close enough to s. Then, for any ball \(B_R\) in \(\mathbb {R}^N\), it follows from Hölder’s inequality that
$$\begin{aligned}&\int ^T_0 \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert ^2_{L^{2}(B_R)} \hbox {d}t \nonumber \\&\quad \leqq |B_R|^{2( 1-\frac{2}{r} ) } \int ^T_0 \Vert \nabla (-\Delta )^{-s}[G(u_{\delta _2})] \Vert ^2_{L^{r}(B_R) } \hbox {d}t \nonumber \\&\quad \lesssim |B_R|^{2(1-\frac{2}{r})} \int ^T_0 \Vert |u_{\delta _2}|^{\gamma -1} u_{\delta _2}\Vert ^2_{\dot{H}^{1-s} (\mathbb {R}^N) } \hbox {d}t . \end{aligned}$$(3.88)By (3.78) and (3.88), we obtain (3.84).
This implies that \(\nabla (-\Delta )^{-s}[G(u_{\delta _2})] \) converges weakly to a function w in \(L^2\left( B_R\times (0,T)\right) \) as \(\delta _2\rightarrow 0\), up to a subsequence.
Moreover, since \(G(u_{\delta _2})\rightarrow G(u)\) for almost everywhere \((x,t)\in \mathbb {R}^N\times (0,T)\), then we obtain (3.83).
-
(iii) The case \(s=\frac{1}{2}\). This case is quite simple.
Indeed, since \(\nabla ^0= \nabla (-\Delta )^{-1/2}= \nabla \mathcal {I}_{1} u \) (the Riesz transform of u), then we have
$$\begin{aligned} \Vert \nabla (-\Delta )^{-1/2} [G(u_{\delta _2})]\Vert ^q_{L^q\left( Q_T \right) } \lesssim \Vert G(u_{\delta _2}) \Vert ^q_{L^q\left( Q_T \right) } \end{aligned}$$for any \(q>1\) such that \(m_2q \geqq 1\). Thanks to Proposition 2, we obtain
$$\begin{aligned} \Vert G(u_{\delta _2}) \Vert ^q_{L^q\left( Q_T \right) } \leqq C , \end{aligned}$$where \(C>0\) is independent of \(\delta _1, \delta _2\). As a result, there is a subsequence of \(\{\nabla (-\Delta )^{-1/2} [G(u_{\delta _2})]\}_{\delta _2>0}\) such that
$$\begin{aligned} \nabla (-\Delta )^{-1/2} [G(u_{\delta _2})] \rightharpoonup \nabla (-\Delta )^{-1/2} [G(u)],\,\, \text {in } L^2 \left( Q_T\right) . \end{aligned}$$(3.89)
Thanks to (3.74), (3.75), (3.80), (3.83) and (3.89), we can pass to the limit as \(\delta _2\rightarrow 0\) in equation (3.72) in order to obtain equation (3.73). In other words, u is a weak solution of (3.71).
Hence, we get the proof of Proposition 7. \(\quad \square \)
Remark 7
We emphasize that the estimates in the proof of Proposition 7 are also independent of \(\delta _1\).
Next, we will pass to the limit as \(\delta _1\rightarrow 0\) in (3.71).
Proposition 8
Let \(u_{\delta _1}\) be a solution of (3.71) . Then, there exists a subsequence of \(\{u_{\delta _1}\}_{\delta _1>0}\), converging to a function u in \(L^2\left( B_R\times (0,T) \right) \) for any \( R>0\).
Furthermore, \(u\in L^1(Q_T)\cap L^\infty (Q_T)\), which is a weak solution of equation (1.1). In addition, we obtain the regularity of \(\text{ div }\left( \Theta (u)\right) \) as follows:
-
If \(s\in [\frac{1}{2},1)\) then
$$\begin{aligned} \text{ div }\left( \Theta (u)\right) \in L^2\left( 0,T; H^{-1}(B_R)\right) . \end{aligned}$$(3.90) -
Otherwise, if \(s\in (0,\frac{1}{2})\) then
$$\begin{aligned} \text{ div }\left( \Theta (u)\right) \in L^p\left( 0,T; W^{-2,p}(\mathbb {R} ^N)\right) , \end{aligned}$$(3.91)for \(p>1\) such that \(\frac{m_2 p}{p-1} \geqq 1\), and \(W^{-2,p}(\mathbb {R}^N)\) is the dual space of \(W^{2,p}(\mathbb {R}^N)\).
Proof
Thanks to Remark 7, we observe that the proof of Proposition 8 can be done by repeating the one of Proposition 7. Thus, it remains to prove \(\delta _1 \Delta u_{\delta _1} \rightarrow 0\) in \(\mathcal {D}' (Q_T)\), as \(\delta _1\rightarrow 0\), (3.90) and (3.91).
We first show that
as \(\delta _1\rightarrow 0\), .
Indeed, for any \(\varphi \in L^2\left( 0,T; H^{1}(\mathbb {R}^N)\right) \) we have from (3.17) and Hölder’s inequality
Hence, (3.92) follows as \(\delta _1\rightarrow 0\).
Next, we prove (3.90). It follows from Hölder’s inequality that
Then,
It follows from (3.84) and (3.89) that
Then, we obtain (3.90).
Next, for any \(p>1\) such that \(m_2 p'\geqq 1\), we have
A combination of the last inequality and (3.80) implies that
Thus, (3.91) follows.
Then, we complete the proof of Lemma 8. \(\quad \square \)
In the case \(m_2>m_1\), we prove that (3.90) holds for any \(s\in (0,1)\).
Proposition 9
Let \(s\in (0,1)\), and let u be a weak solution of (1.1). Assume that \(m_2>m_1\). Then, for any ball \(B_R\) it holds that
Proof
By the same argument as in (3.77), we also obtain
for \(q>1\). Let us take \(q=1+ m_2-m_1\). It follows from (3.94) that
Thus,
Now, we have from the Plancherel theorem that
Applying Hölder’s inequality, we obtain
On the other hand, it follows from the Plancherel theorem that
Moreover, we apply the Riesz potential estimate and Hölder’s inequality to get
Note that \(\frac{2N}{N+2s}>1\) since \(s\in (0,1)\) and \(N\geqq 2\).
From (3.99) and (3.98) we obtain
A combination of (3.96), (3.97) and (3.100) yields
where \(C=C(R, u_0, N, s, m_1)>0\). Then,
Thus, the conclusion follows from the last inequality and (3.95) . \(\quad \square \)
4 Decay Estimates and the Finite Time Extinction of Solution
In this part, we study some decay estimates and the finite time extinction of solution u. We start with the case \(f=0\) by proving Theorem 2.
Proof of Theorem 2
By technical reasons, we consider first the case \(p>1\). It follows from (3.18) and after passing to the limit as \(\varepsilon , \kappa , \nu , \delta _2\rightarrow 0\) that
Thanks to Lemma 5, we obtain
with \(\theta _0=\frac{m_1+m_2+ p-1}{2}=\frac{\beta _0 + p}{2}\),
By Sobolev embedding, we have
with \(C=C(N,s,2)\), and \(2^\star =\frac{2N}{N-2(1-s)}=\frac{2}{\alpha _0}\).
In order to use an iteration method, let us put \(q_0=p\). A combination of the two last inequalities leads to
with \(q_1=2^\star \theta _0\). Integrating both sides of (4.2) on \((s,t)\subset \subset (0,T)\) yields
It follows from (4.1) that \(\Vert u(t)\Vert _{L^{q_0}} \) is nonincreasing with respect to t. Then, (4.3) gives
for every \(0<s<t<T\). It is important to note that the constant C in (4.4) will not change step by step, since we are going to use an iteration method starting from here.
Now, let us set
and states that \(\theta _0, q_0\) are as above. Here, we note that the condition \(m_1+m_2>1-\frac{2q_0(1-s)}{N}\) is equivalent to \(q_1>q_0\). Thus, by induction, we observe that the sequence \(\{ q_{n} \}_{n\geqq 0}\) is increasing.
Let us take \(t=t_{n+1}, s=t_n\), and replace \(q_0\) by \(q_n\) in (4.4). Then, we obtain
By induction, we get
It is not difficult to verify that
Next, by using (4.6), we obtain
Similarly, we also have
and
After that, let us put
Let us show that \(Z_n\) is convergent as \(n\rightarrow \infty \). Indeed, we consider the power series
Obviously, the radius of convergence of \(S_n(s)\) is 1. Thus, \(S_n (\alpha _0) \) converges absolutely to a real number \(\lambda _0\) as \(n\rightarrow \infty \).
On the other hand, we note that
It follows then from (4.6) that
Then,
Similarly, there is a real positive number \(\zeta _0\) such that
A combination of (4.5), (4.6), (4.7), (4.8), (4.9), (4.11), and (4.12) implies that there is a constant \(C=C(N,s,q_0,m_1,m_2)>0\) such that
This puts an end to the proof of Theorem 2 when \(p>1\).
Next, we prove \(L^1\) decay estimate. To do this, we first prove an estimate of the decay \(L^1\)-\(L^q\) in the following lemma:
Lemma 9
Let \(s\in (0,1)\), and \(m_1, m_2>0\) be such that \(m_1+m_2>\alpha _0\). Assume that \(u_0\in L^{1}(\mathbb {R}^N)\). Then, for any \(q>1\) it holds that
with \(C=C(N,s, m_1, m_2, q)>0\).
Proof
Thanks to the Interpolation Inequality and the monotonicity of \(\Vert u(t)\Vert _{L^1}\) for \(t>0\), it suffices to prove that (4.14) is true for any \(q>1\) large enough. We mimic the proof of Theorem 2 above by considering \(q=q_0\). Let us recall here \(q_1= \frac{\beta _0 +q_0}{\alpha _0}\). Note that \(q_1>q_0\) since \(q_0\) is large enough.
Then, we apply the Interpolation Inequality to obtain
with \(\theta =\frac{\frac{1}{q_0}-\frac{1}{q_1} }{1-\frac{1}{q_1}}\). Since \(\Vert u(t)\Vert _{L^1}\) is nonincreasing for \(t\geqq 0\), we then get
It follows from (4.2) and the last inequality that
with \(C=C(N, s, q_0)>0\), and \(y(t)=\Vert u(t)\Vert ^{q_0}_{L^{q_0}}\).
Note that \(1-\frac{\alpha _0 q_1}{(1-\theta )q_0}<0\) since \(q_0\) is large enough. Then, solving the OD inequality yields
with \(C=C(N,s,q_0, m_1, m_2)>0\).
Thus,
This completes the proof of Lemma 9. \(\quad \square \)
Now, let us prove Theorem 2 for \(p=1\). In fact, we have, from (4.13), that
for any \(\tau \in (0,t)\) and for any \(q_0\) large enough.
Thanks to (4.14), we obtain
Then, the conclusion follows from (4.16) and (4.17) with \(\tau =\frac{t}{2}\). And the proof of Theorem 2 is complete. \(\quad \square \)
Finally, let us prove Theorem 3.
Proof of Theorem 3
We recall here \(q_0, q_1\) as in the proof of Theorem 2. Then, we can mimic the proof of Lemma 9 in order to obtain, as in (4.15), that
with \(y(t)=\Vert u(t) \Vert ^{q_0}_{L^{q_0}}\).
Now, we shall show that there exists a time \(\tau _0>0\) such that \(y(\tau _0)=0\). If this is done, then the conclusion follows from the monotonicity of y(t) by (4.18). Assume by contradiction that \(y(t)>0\), for any \(t>0\). Then, we solve the indicated ODE to obtain
Therefore,
By the fact \(1-\frac{\alpha _0 q_1}{(1-\theta ) q_0}>0\), (4.19) leads to a contradiction as \(t\rightarrow \infty \).
Thus, we obtain the proof of Theorem 3. \(\quad \square \)
Remark 8
By (4.19), we can estimate the extinction time of u, denoted as \(\tau _0\) satisfying
Remark 9
Some of the results of this paper remain valid for the case of a bounded domain and homogeneous Dirichlet boundary conditions. Moreover function f may be given through a potential as in the case of nonlocal Schrödinger equation, such as \(f(x,t)=-V(x) u(x,t)\); see [19]. To end this paper, we would like to refer to [1] for the study of the energy method in proving the complete quenching of solutions and the free boundary of solutions of nonlinear evolution equations.
Now we shall consider the case of \(f\ne 0\). In fact, the finite time extinction phenomenon also appears in problem (1.1) when \(f\ne 0\) and f extincts in a finite time \(T_{f}>0\) (i.e: \(f(x,t)=0\) for \(t\geqq T_{f}\), and \(x\in \mathbb {R}^N\)). By assuming, for simplicity, that
it is easy to adapt the proof of Theorem 3 to conclude that
with \(y(t)=\Vert u(t)\Vert _{L^{q_{0}}}^{q_{0}}\) and
for some positive constants \(K_{1}\) and \(K_{2}.\) Thus, we get the existence of a finite extinction time \(\tau _{0}\geqq \)\(T_{f}\) for the solution u of problem (1.1) by repeating the arguments of Theorem 4 starting with the initial datum \(u(x,T_{f}).\)
A less intuitive fact is that for certain source functions, \( f(t)\ne 0\), with a finite extinction time \(T_{f}>0\), the resulting extinction time \(\tau _{0}\) of the solution u let such that \(\tau _{0}=\)\( T_{f}\). Such behavior was considered in the monograph [1] (see Theorem 2.1 of Chapter 2) for the case of local problems. As many other free boundary problems, this phenomenon requires a suitable balance between the domain (here the interval \((0,T_{f})\)) and the datum \( \Vert u_0\Vert _{L^{q_{0}}}\), with a suitable decay of the right hand side (here given by the decay of g(t) around \((t-T_{f})_{+}\)).
Proposition 10
Let \(s\in (0,1)\), and let \(m_1, m_2>0\) be such that \(m_1+m_2 < \alpha _0=\frac{N-2(1-s)}{N}\). Assume that \(\Vert u_0\Vert _{L^1( \mathbb {R}^N)}+\Vert u_0\Vert _{L^\infty (\mathbb {R}^N)}\) is small enough. Let \(\nu _0\) satisfy
Suppose that \(f\in L^{1}(Q_{T})\cap L^{\infty }(Q_{T})\) and there exists a finite time \(T_f>0\) such that
and for some \(\varepsilon >0\) small enough. Then the finite extinction time of the solution u coincides with the extinction time of the source term f , that is \(\tau _{0}=T_{f}\).
Proof
Let us set
Note that \(q_0 \geqq 2\) since \(\nu _0> 1+m_1+m_2-\alpha _0\). By a simple calculation, we have
with \(q_1=\frac{\beta _0+q_0}{\alpha _0}\), and \(\theta =\frac{\frac{1}{q_0}-\frac{1}{q_1}}{1-\frac{1}{q_1}}\). We also emphasize that \(q_1>q_0\) since \(\nu _0>\alpha _0\).
In a similar way to the proof of (4.18), and thanks to the assumption (4.21), we observe that \(y(t)=\Vert u(t)\Vert _{L^{q_{0}}}^{q_{0}}\) satisfies the ordinary differential inequality
for some positive constant K. However, it is easy to see that the function
is a supersolution of problem (4.22) once we assume the following condition on the data:
We note that (4.23) occurs since \(u_0\) is small and \(\varepsilon >0\) is also small enough, and it depends on \(u_0\). Then, by applying the comparison principle for nonnegative solutions of the ODE associated to (4.22), we get
which implies that the extinction time of y(t) coincides with \(T_{f}\). \(\quad \square \)
Remark 10
Notice that Theorem 3 extends to the case of the nonlocal problem (1.1) the result by Bénilan and Crandall [2] when we take \(s=0\) and \(m:=m_{1}+m_{2}\).
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Acknowledgements
The first author would like to thank the Vietnam Institute of Advanced Study in Mathematics (VIASM) for their warm hospitality during his visiting time. The research of Jesus Ildefonso Díaz was partially supported by Project MTM2017-85449-P of the Spanish Ministerio de Ciencia e Innovación and as a member of the Research Group MOMAT (Ref. 910480) of the UCM.
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Appendix
Appendix
Lemma 10
Let \(s\in (0,1)\). For any \(\varepsilon >0\), we have
Proof
It is known that for any \(u\in \mathcal {S}(\mathbb {R}^N)\) (the Schwartz space), \(\mathcal {F} \{(-\Delta )^s\}\) can be considered as a multiplier of \(\mathcal {F} \{(-\Delta )^s u \}\), that is,
We have
Taking the Fourier transform yields
This implies that
Similarly, we also have
Therefore,
Then the conclusion of Lemma 10 follows from (5.1) and (5.2). \(\quad \square \)
Next, we have the following embedding results:
Lemma 11
Let \(\alpha \in (0,1)\), \(N\geqq 1\), and \(p\geqq 1\). Then, we have
Proof
We have
Now we consider \(\mathbf {I}_1\). Applying Young’s inequality and Hölder’s inequality yields
Next, we apply Young’s inequality to get
A combination of (5.4) and (5.5) implies
The last inequality holds for any \(\lambda >0\), so we obtain
By (5.3) and (5.6), we complete the proof of Lemma 11. \(\quad \square \)
Lemma 12
Let \(\theta \in (0,1)\), and \(N\geqq 1\). Let \( \alpha _1, \alpha _2 \in (0,1)\) be such that \(\alpha _1 < \alpha _2\theta \). Assume that \(\Gamma \) is a \(\theta \)-Hölder continuous function on \(\mathbb {R }\). Then we have
where
Remark 11
Note that it follows from (5.7) that \(r>2\).
Proof
The proof of Lemma 12 can be found in [3, Lemma 6.6] . \(\quad \square \)
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Dao, N.A., Díaz, J.I. Energy and Large Time Estimates for Nonlinear Porous Medium Flow with Nonlocal Pressure in \(\mathbb {R}^N\). Arch Rational Mech Anal 238, 299–345 (2020). https://doi.org/10.1007/s00205-020-01543-1
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DOI: https://doi.org/10.1007/s00205-020-01543-1