# Critical extinction exponents for a nonlocal reaction-diffusion equation with nonlocal source and interior absorption

- Bing Gao
^{1}and - Jiashan Zheng
^{1}Email author

**2014**:19

https://doi.org/10.1186/1687-1847-2014-19

© Gao and Zheng; licensee Springer. 2014

**Received: **10 October 2013

**Accepted: **23 December 2013

**Published: **16 January 2014

## Abstract

This paper is concerned with a nonlocal reaction-diffusion equation with nonlocal source and interior absorption ${u}_{t}={\int}_{{\mathbb{R}}^{N}}J(x-y)(u(y,t)-u(x,t))\phantom{\rule{0.2em}{0ex}}dy+\lambda {\int}_{\mathrm{\Omega}}{u}^{q}\phantom{\rule{0.2em}{0ex}}dx-{u}^{p}$, $x\in \mathrm{\Omega}$, $t>0$, $u(x,t)=0$, $x\notin \mathrm{\Omega}$, $t\ge 0$, $u(x,0)={u}_{0}(x)$, $x\in \mathrm{\Omega}$. We investigate the critical extinction exponents for the problem based on some adequate supersolutions and subsolutions.

**MSC:**35K57, 35B33, 35K10.

## Keywords

## 1 Introduction

where $J:{\mathbb{R}}^{N}\to \mathbb{R}$ is a nonnegative, smooth, symmetric radially function with ${\int}_{{\mathbb{R}}^{N}}J(z)\phantom{\rule{0.2em}{0ex}}dz=1$ and supported in the unitary ball, $\lambda ,p,q>0$. We assume that ${u}_{0}\in C(\mathrm{\Omega})$ is a nonnegative function.

*x*at time

*t*, and $J(x-y)$ is thought of as the probability distribution of jumping from location

*y*to location

*x*, then ${\int}_{{\mathbb{R}}^{N}}J(x-y)u(y,t)\phantom{\rule{0.2em}{0ex}}dy$ and $-u(x,t)=-{\int}_{{\mathbb{R}}^{N}}J(x-y)u(x,t)\phantom{\rule{0.2em}{0ex}}dy$ is the rate at which individuals are arriving at position

*x*from all other places and at which they are leaving location

*x*to travel to all other sites, respectively. It is well known that equation (1.2) shares many properties with the classical heat equation, ${u}_{t}=\mathrm{\u25b3}u$, such as the bounded stationary solutions and the maximum principle [6]. In the last few years, a lot of works have been devoted to the study of properties of solutions to parabolic problems involving nonlocal terms. Especially, García-Melián and Rossi [7] discussed the existence of a critical exponent of Fujita type for the nonlocal diffusion problem with local source. Zhang and Wang [8] studied the critical exponent for the nonlocal diffusion equation

with $p,q\in (0,1)$ and $k,d>0$ and showed that $q=p$ is the critical extinction exponent by invoking the regularizing effect. In this paper under the appropriate hypotheses $p,q>0$, we discuss problem (1.1) and obtain the extinction condition by using the principal eigenvalue of the nonlocal heat equation, and thus avoid using the regularizing effect, since there is no regularizing effect in general [18]. It is noted that our approach can be adopted to deal with the blow-up behavior of solutions of nonlocal reaction-diffusion equations with nonlocal source or local source, which was considered in [7, 19].

Now our main results can be stated as follows.

## 2 Main results

**Theorem 2.1**(1)

*If*$q=1$,

*then the solution of problem*(1.1)

*vanishes in infinite time for any nonnegative initial data provided that*$\lambda >0$

*is appropriately small*.

- (2)
*If*$q>1$,*then the solution of problem*(1.1)*vanishes in infinite time for any appropriately small initial data*.

**Theorem 2.2**(1)

*If*$q=p<1$,

*then the solution of problem*(1.1)

*vanishes in finite time for any nonnegative initial data provided that*$\lambda >0$

*is appropriately small*.

- (2)
*If*$p<q<1$,*then the solution of problem*(1.1)*vanishes in finite time for any conveniently small initial data*.

**Remark 1**(1) The small condition on initial data ${u}_{0}$ in Theorem 2.1 and Theorem 2.2 can be removed if $\lambda >0$ is sufficiently small.

- (2)
That $u(x,t)$ vanishes in infinite time means that ${lim}_{t\to +\mathrm{\infty}}u(x,t)=0$ for any $x\in \mathrm{\Omega}$.

- (3)
That $u(x,t)$ vanishes in finite time means that there exists $0<{T}^{\ast}<+\mathrm{\infty}$, such that $u(x,t)=0$ for any $x\in \mathrm{\Omega}$ and $t>{T}^{\ast}$.

**Theorem 2.3**(Nonextinction)

- (1)
*If*$q=p<1$,*then problem*(1.1)*admits at least one nonextinction solution for any nonnegative initial data provided that*$\lambda >0$*is appropriately large*. - (2)
*If*$q=1<p$,*then problem*(1.1)*admits at least one nonextinction solution for any nonnegative initial data provided that**λ**is appropriately large*. - (3)
*If*$q<min\{1,p\}$,*then problem*(1.1)*admits at least one nonextinction solution for any nonnegative initial data*. - (4)
*If*$q=p=1$,*then problem*(1.1)*admits at least one nonextinction solution for any nonnegative initial data provided that*$\lambda >0$*is sufficiently large*.

### Preliminary lemmas

Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. As for the proofs of these lemmas, we will not repeat them again.

Applying almost exactly the same arguments as in the proof of Lemma 5 in [21], we conclude to the following lemma.

**Lemma 2.1**

*Let*$y(t)$

*be a solution of the following problem*:

*where* $\alpha ,\beta ,\gamma >0$ *and* $0<q<min\{p,1\}$. *Then the above ODE problem has at least one non*-*constant solution*.

Next, our aim is to prove the local existence of solutions to equation (1.1) and the validity of the comparison principle. First, we give the definition of supersolution and subsolution.

**Definition 2.1**A nonnegative function

where ${Q}_{T}:=\mathrm{\Omega}\times (0,T)$. The subsolution is defined similarly by reversing the inequalities. Furthermore, if *u* is a supersolution as well as a subsolution, then we call it a solution of problem (1.1).

The existence of the solution of problem (1.1) will be obtained via the successive approximation which comes from [22].

**Lemma 2.2** *Let* $0\le {u}_{0}\in C(\overline{\mathrm{\Omega}})$. *Then there exists* $T=T(\lambda ,p,q)>0$, *such that problem* (1.1) *has nonnegative solutions*.

*Proof*Let

with $0\le {u}_{m}(x,t)\le {\parallel {u}_{0}\parallel}_{{L}^{\mathrm{\infty}}}+1$ for all most $(x,t)\in {Q}_{T}$. Then it follows from the Lebesgue dominated convergence theorem that $u(x,t)$ is the solution of problem (1.1). □

In the following, we conclude that a comparison principle holds for solutions to problem (1.1).

**Lemma 2.3** *Let* $\overline{u}$, $\underline{u}$ *be the supersolution and the subsolution to equation* (1.1), *respectively*. *If either* $q\ge 1$ *and* $\overline{u}$ *is upper bounded or* $0<q<1$ *and* $\underline{u}$ *has a positive lower bound*, *then* $\underline{u}(x,t)\le \overline{u}(x,t)$ *in* ${Q}_{T}$.

*Proof*Let $v=\underline{u}-\overline{u}$. Then due to the Definition 2.1, we have

*M*depends only on $\overline{u}$, where ${s}_{+}=max\{s,0\}$ and $M>0$. It then follows from Gronwall’s inequality that

which implies that $\underline{u}(x,t)\le \overline{u}(x,t)$ in ${Q}_{T}$. The assertion can be proved similarly for the case $0<q<1$ and $\underline{u}$ has a positive lower bound. Thus the proof of this lemma is completed. □

Once the existence of the solution to problem (1.1) and the comparison principle are ensured, we begin to analyze the extinction exponents for nonnegative solutions. As a first step we discuss the infinite time extinction of the solution.

*Proof of Theorem 2.1* The proof can be divided into two steps:

Therefore, applying Lemma 2.3 to equation (1.1) in ${Q}_{T}$, we have $u(x,t)\le v(x,t)$ for $(x,t)\in {Q}_{T}$, which implies ${lim}_{T\to +\mathrm{\infty}}u(x,T)=0$. Hence the solution of equation (1.1) vanishes in infinite time provided that $\lambda <\frac{{\lambda}_{1}m}{{\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx}$.

*Q*, we obtain $u(x,t)\le v(x,t)$ for $(x,t)\in Q$, which implies that $u(x,t)\le AM$. Therefore $u(x,t)$ satisfies

and then by Step I, we end up with that the solution $u(x,t)$ of equation (1.1) vanishes in infinite time. The proof of this theorem is completed. □

*Proof of Theorem 2.2* The proof is similar to that of Theorem 2.1, so we sketch it briefly here. We will prove the theorem in two cases.

Thus, thanks to Lemma 2.3, we derive $u(x,t)\le w(x,t)$ ($(x,t)\in {Q}_{T}$), for any fixed $T<{T}^{\ast}$. Therefore, $u(x,T)\le w(x,T)$, which, together with the arbitrariness of $T<{T}^{\ast}$ and $w(x,{T}^{\ast})=0$ implies that $u(x,{T}^{\ast})=0$. Furthermore, setting $\tilde{u}(x,t)=u(x,t+{T}^{\ast})$, then $\tilde{u}(x,t)$ satisfies equation (1.1). According to the above proof, we claim that $\tilde{u}(x,t)\le w(x,t)$ with any ${g}_{0}>0$. Now, by virtue of the relation of the extinction time ${T}^{\ast}$ of $w(x,t)$ to ${g}_{0}$, we finally conclude that $\tilde{u}(x,t)=0$ for any $t>0$, namely $u(x,t)=0$ for all $t\ge {T}^{\ast}$.

According to the above results, the solution $u(x,t)$ of equation (1.1) vanishes in finite time. This completes the proof of Theorem 2.2. □

*Proof of Theorem 2.3* The proof can be divided into four cases.

which implies *w* is a subsolution of problem (1.1). Therefore problem (1.1) admits a solution $u(x,t)$ satisfying $u(x,t)\ge w(x,t)$, which, combined with $w(x,t)>0$ ($\mathrm{\Omega}\times (0,+\mathrm{\infty})$) implies that $u(x,t)$ is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that $\lambda >0$ is appropriately large.

Since $p>1$ and $\lambda {\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx-{\lambda}_{1}M>0$, we conclude that $g(t)$ is a nondecreasing and $g(t)\le {(\frac{\frac{\lambda {\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx}{M}-{\lambda}_{1}}{{m}^{p-1}})}^{\frac{1}{p-1}}$. Let $w(x,t)=g(t)\psi (x)$. Then we can easily derive $u(x,t)\ge w(x,t)$. Therefore, $u(x,t)$ is a nonextinction solution of equation (1.1) for any nonnegative initial data provided that $\lambda >0$ is appropriately large.

Applying Lemma 2.1 to equation (2.6), we have $g(t)>0$ ($t>0$). Then the same argument as in the derivation of Case I shows that $u(x,t)$ is a nonextinction solution of equation (1.1) for any nonnegative initial data.

Case IV: If $q=p=1$ and $\lambda {\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx-({\lambda}_{1}+1)M>0$, employing exactly the same arguments as in the proof of Case I, we finally conclude the result. □

## Declarations

## Authors’ Affiliations

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