Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption

We consider the initial Dirichlet boundary value problem for a class of porous medium equations with nonlocal source and strong absorption terms

(1)

(2)

(3)

where $0<m,k<1$, $d,\lambda ,\beta ,q>0$, $\mathrm{\Omega }\subset R^N$ ($N\ge 1$) is a bounded domain with smooth boundary, and $u_0(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })\cap W_0^{1,2}(\mathrm{\Omega })$ is a nonnegative function. The symbols ${\parallel \cdot \parallel }_p$ and ${\parallel \cdot \parallel }_{1,p}$, where $p\ge 1$, denote $L^p(\mathrm{\Omega })$- and $W^{1,p}(\mathrm{\Omega })$-norm, respectively, and $|\mathrm{\Omega }|$ denotes the measure of Ω.

Equation (1) describes the fast diffusion of concentration of some Newtonian fluids through a porous medium or the density of some biological species in many physical phenomena and biological species theories, while nonlocal source and absorption terms cooperate and interact with each other during the diffusion. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1–3]. In the nonlinear diffusion theory, obvious differences exist among the situations of slow ($m>1$), fast ($0<m<1$), and linear ($m=1$) diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situations, whereas an infinite speed propagation exists in the fast diffusion situation.

subject to (2) and (3). They obtained some conditions for the extinction and non-extinction of solutions to equation (4) and decay estimates by the upper and lower solutions method. On extinctions of solutions to the p-Laplacian equations or the doubly degenerate equations, we refer readers to [13, 14] and the references therein.

and obtained sufficient conditions for the extinction and non-extinction of solutions to that equation. Thereafter, Fang and Li [16] extended their results to the doubly degenerate equation in the whole dimensional space.

For equation (1) with $\beta =0$, $q>0$, and $N>2$, Han and Gao [17] showed that $q=m$ is the critical exponent for the occurrence of extinction or non-extinction. When $m=1$, $q>0$, and $N>2$, the conditions for the extinction and non-extinction of solutions and corresponding decay estimates were obtained (cf. [18]). Recently, Fang and Xu [19] considered equation (1) with $k=1$, when the diffusion term was replaced with a p-Laplacian operator in the whole dimensional space, and showed that the extinction of the weak solution is determined by the competition of two nonlinear terms. They also obtained the exponential decay estimates which depend on the initial data, coefficients, and domains. The extinctions of solutions to equation (1) with nonlocal source terms do not depend on the first eigenvalue of the corresponding operator, which is different from the case of local source terms. The extinction and decay estimates for solutions to the nonlocal fast diffusion equations with nonzero coefficients and strong absorption terms, like equation (1), are still being investigated.

Motivated by the above works, we investigate whether the existence of strong absorption can change extinction behaviors for solutions to problem (1)-(3) in the whole dimensional space. The main tools we use are the integral estimate method and the Gagliardo-Nirenberg inequality. This technique has a wide application, especially for equations that do not satisfy the maximum principle (cf. [20]). Our goals are to show that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when $0<k\le q<1$ and to find the decay estimates depending on the initial data, coefficients, and domains.

Our paper is organized as follows. In Section 2, we give preliminary knowledge including lemmas that are required in the proofs of our results and present the proofs for the results in Section 3.

A function u is called a locally weak solution of problem (1)-(3) if it is both a subsolution and a supersolution for some $T>0$.

Remark 1 The existence and uniqueness of locally nonnegative solutions in time to problem (1)-(3) can be obtained by the standard parabolic regular theory that can be applied to get suitable estimates in the standard limiting process (cf. [2, 21, 22]). The proof is similar to the ones in the cited references, and so it is omitted here.

where $T_{\ast }=\frac{y^{1-k}(0)}{\alpha (1-k)}$.

Proof

and using the comparison principle, one can easily obtain the result. □

Lemma 2 (The Gagliardo-Nirenberg inequality) [23]

where C is a constant depending on N, m, r, j, k, and q such that $\frac{1}{q}=\frac{j}{N}+\theta (\frac{1}{m}-\frac{k}{N})+\frac{1-\theta }{r}$ and $\theta \in [0,1)$. While if $m<\frac{N}{k-j}$, then $q\in [\frac{Nr}{N+rj},\frac{Nm}{N-(k-j)m}]$, and if $m\ge \frac{N}{k-j}$, then $q\in [\frac{Nr}{N+rj},+\mathrm{\infty }]$.

In this section, we give some extinction properties of nonnegative nontrivial weak solutions of problem (1)-(3) stated in the following theorems. The corresponding decay estimates to the solutions will be presented in the proofs of the theorems for brief expressions instead of in the statements.

Theorem 1 Suppose that $0<k<1$ and $0<m=q<1$. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that either $|\mathrm{\Omega }|$ or λ is sufficiently small.

where ${\theta }_1=\frac{m+1}{2}(\frac{1}{k+1}-\frac{1}{2}){(\frac{1}{N}-\frac{1}{2}+\frac{m+1}{2}\frac{1}{k+1})}^{-1}=\frac{N(1-k)(m+1)}{2[2(k+1)+N(m-k)]}$. Since $N=1\text{ or }2$, and $0<k<1$, it can be easily seen that $0<{\theta }_1<1$.

Setting $C_1=\frac{C{(N,m,k)}^{-k_1}\beta }{C({\eta }_1)}$, we have $\frac{d}{dt}{\parallel u\parallel }_2+C_1{\parallel u\parallel }_2^{k_1-1}\le 0$.

where $T_1=\frac{{\parallel u_0\parallel }_2^{2-k_1}}{C_1(2-k_1)}$, which give the decay estimates in finite time for $N=1\text{ or }2$.

where $T_2=\frac{{\parallel u_0\parallel }_{s+1}^{s+1-k_2}}{C_2(s+1-k_2)}$, which give the decay estimates in finite time for $N>2$ such that $\frac{N-2}{N+2}\le m<1$.

where $T_2^{\ast }=\frac{C({\eta }_2){\parallel u_0\parallel }_{s+1}^{s+1-k_2}}{C{(N,k,s)}^{-k_2}\beta (s+1-k_2)}$. □

Theorem 2 If $0<k<1$, then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time provided that $u_0$, $|\mathrm{\Omega }|$ or λ is sufficiently small, and $q>\frac{2k(s+1)+N(m-k)}{2(s+1)+N(m-k)}$, where if $N=1\text{ or }2$, then $s=1$, and if $N>2$, then $s>m$.

By (10) and (18), and using Hölder’s inequality, we get the inequality

where $T_3=\frac{{\parallel u_0\parallel }_2^{2-k_1}}{C_3(2-k_1)}$.

where $T_4=\frac{{\parallel u_0\parallel }_{s+1}^{s+1-k_2}}{C_4(s+1-k_2)}$.

Since $s>m$, we have $2(s+1)>2m+2$, and hence, if $k\ge m$, then $q>k_2-s\ge m$.

where $T_4^{\ast }=\frac{C({\eta }_2){\parallel u_0\parallel }_{s+1}^{s+1-k_2}}{C{(N,k,s)}^{-k_2}\beta (s+1-k_2)}$. □

Remark 2 Since the Sobolev embedding inequality cannot be used in the proof of Theorem 2, it is not necessary to consider the cases that $\frac{N-2}{N+2}\le m<1$ and $0<m<\frac{N-2}{N+2}$, when $N>2$. In addition, if $k\ge m$, the conditions in Theorem 2 imply that $q>m$.

Theorem 3 Suppose that $0<k<1$ and $m>q\ge k$. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that β is sufficiently large.