 Research
 Open Access
 Published:
Critical extinction exponents for a nonlocal reactiondiffusion equation with nonlocal source and interior absorption
Advances in Difference Equations volume 2014, Article number: 19 (2014)
Abstract
This paper is concerned with a nonlocal reactiondiffusion equation with nonlocal source and interior absorption {u}_{t}={\int}_{{\mathbb{R}}^{N}}J(xy)(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.
1 Introduction
Our goal is to study the critical extinction exponents of the nonlocal heat equation with nonlocal source and interior absorption, namely,
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.
Since the longrang effects are taken into account, nonlocal diffusion equations of the form
have been widely used to model the diffusion processes (see [1–6] and references therein). More precisely, as stated in [6], if u(x,t) is thought of as the density of a species at the point x at time t, and J(xy) is thought of as the probability distribution of jumping from location y to location x, then {\int}_{{\mathbb{R}}^{N}}J(xy)u(y,t)\phantom{\rule{0.2em}{0ex}}dy and u(x,t)={\int}_{{\mathbb{R}}^{N}}J(xy)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, GarcíaMelián 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>1, \beta >0 and obtained the critical exponent {p}_{\beta}^{\ast}. Recently, Fang and Xu [9] investigated the extinction behavior of solutions for the homogeneous Dirichlet boundary value problem of the nonNewtonian filtration equation with nonlocal sources. More recently, Antontsev and Shmarev [10] discussed the behavior of energy solutions of the homogeneous Dirichlet problem for the anisotropic doubly degenerate parabolic equation
with m(x,t)>0, {p}_{i}(x,t)>1 and \sigma (x,t)>1. They derived the sufficient conditions of the finite time blowup or vanishing and established the decay rates as t\to \mathrm{\infty}. More results on the extinction for the degenerate parabolic equations have also been obtained by many researchers, and we may refer to [11–16] and the references therein. We point out that Liu [17] investigated the extinction properties of solutions for the homogeneous Dirichlet boundary value problem of the nonlocal reactiondiffusion 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 blowup behavior of solutions of nonlocal reactiondiffusion equations with nonlocal source or local source, which was considered in [7, 19].
Motivated by the above works, the purpose of this paper is to analyze the extinction exponent for problem (1.1), that is, we want to show that problem (1.1) shares many important properties with the corresponding local reactiondiffusion system,
such as the extinction condition [17]. Through the main points, we see that there exists a critical curvilinear line {q}^{\ast}=min(p,1) such that the (q,p)parameter plane is divided into three parts, with the bottom part corresponding to a nonextinction solution and the top part of the line corresponding to all the infinite time extinctions or the finite time extinction. Moreover, there exists a critical point on this line such that the line is also divided into three parts, which exhibits different features of extinction phenomena (see Figure 1).
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.
Let \psi (x) satisfy
Using results in [7, 20] and \psi (x)>0 in \overline{\mathrm{\Omega}}, we can always assume
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 nonconstant 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
is a supersolution of problem (1.1) if it is a supersolution of problem (1.1), which satisfies
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
Then for any m\ge 1, we consider the following successive approximation problem:
Applying almost exactly the same arguments as in the proof of Theorem 1.1 in [22], we derive that equation (2.3) possesses a unique solution {u}_{m}\in C(0,T;{L}^{1}(\mathrm{\Omega}))\cap {W}^{1,1}(0,T;{L}^{1}(\mathrm{\Omega})). Now we turn to proving that
In fact, if m=1, then it is easier to see that {u}_{0}(x,t)={\parallel {u}_{0}\parallel}_{{L}^{\mathrm{\infty}}}+Rt and 0 are a supersolution and a subsolution of equation (2.3), respectively. Then by the comparison principle, we have 0\le {u}_{1}(x,t)\le {u}_{0}(x,t). The fact that 0\le {u}_{m}(x,t)\le {u}_{m1}(x,t)\le {u}_{0}(x,t) in {Q}_{T} can be shown by mathematical induction. Therefore, u(x,t):={lim}_{m\to \mathrm{\infty}}{u}_{m}(x,t) is the solution of equation (1.1). In fact {u}_{m}(x,t) is the solution of the following problem:
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
If q>1 and \overline{u} is bounded, multiplying equation (2.4) by {(\underline{u}\overline{u})}_{+} and integrating it over Ω, we derive
and 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:
Step I: q=1 with \lambda <\frac{{\lambda}_{1}m}{{\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx}. Let v(x,t)=f(t)\psi (x), where f(t) satisfies
Obviously, f(t) is nonincreasing and {lim}_{t\to +\mathrm{\infty}}f(t)=0. Then it can be observed that v(x,t) is the supersolution of equation (1.1) provided that {f}_{0}\ge \frac{{\parallel {u}_{0}\parallel}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}}{m}. To this end, due to q=1, we obtain
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}.
Step II: q>1. Assume that u(x,t) is the solution of equation (1.1) with the initial datum {u}_{0}(x). Let v(x,t)=A\psi (x) with
Then v(x,t) is a supersolution of equation (1.1) provided that {\parallel {u}_{0}\parallel}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}\le Am. Invoking Lemma 2.3 to equation (1.1) in 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.
Case I: If q=p<1 and \lambda <\frac{{m}^{p}}{{\int}_{\mathrm{\Omega}}{\psi}^{q}(x)\phantom{\rule{0.2em}{0ex}}dx}. Let w(x,t)=g(t)\psi (x), where g(t) satisfies the following ODE problem:
Due to p<1, g(t) is nonincreasing and g(t)=0 for all t\ge {T}^{\ast}=\frac{M}{({m}^{p}\lambda {\int}_{\mathrm{\Omega}}{\psi}^{p}(x)\phantom{\rule{0.2em}{0ex}}dx)(1p)}{g}_{0}^{1p}. Hence, we can infer that w(x,t) is the supersolution of equation (1.1) provided that {g}_{0}\ge \frac{{\parallel {u}_{0}\parallel}_{{L}^{\mathrm{\infty}}(\mathrm{\Omega})}}{m}. In fact, with the help of q=p, we readily find that
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}.
Case II: Set 0<p<q<1. Suppose that u(x,t) is the solution of equation (1.1) with the initial datum {u}_{0}(x) and
Let w(x,t)=B\psi (x). By the arguments as those in the proof of Theorem 2.1, we get
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.
Case I: In the case q=p<1 with \lambda {\int}_{\mathrm{\Omega}}{\psi}^{p}(x)\phantom{\rule{0.2em}{0ex}}dxM{m}^{p1}>0, we shall prove that problem (1.1) admits at least one nonextinction solution for any nonnegative initial data by constructing a suitable subsolution of equation (1.1). Let w(x,t)=g(t)\psi (x), where g(t) satisfies
With the help of p<1 and \lambda {\int}_{\mathrm{\Omega}}{\psi}^{p}(x)\phantom{\rule{0.2em}{0ex}}dxM{m}^{p1}>0, we derive that g(t) is nondecreasing and g(t)\le {(\frac{\frac{\lambda {\int}_{\mathrm{\Omega}}{\psi}^{p}(x)\phantom{\rule{0.2em}{0ex}}dx}{M}{m}^{p1}}{{\lambda}_{1}})}^{\frac{1}{1p}}. Simple calculations show that
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.
Case II: Suppose that q=1<p, \lambda {\int}_{\mathrm{\Omega}}\psi (x)\phantom{\rule{0.2em}{0ex}}dx{\lambda}_{1}M>0 and g(t) is the solution of the following ODE problem:
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}^{p1}})}^{\frac{1}{p1}}. 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.
Case III: Suppose that q<min\{p,1\} and let w(x,t)=g(t)\psi (x), where g(t) is given by
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. □
References
 1.
Bates P, Chmaj A: An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Stat. Phys. 1999, 95: 11191139. 10.1023/A:1004514803625
 2.
Bates P, Fife P, Ren X, Wang X: Travelling waves in a convolution model for phase transitions. Arch. Ration. Mech. Anal. 1997, 138: 105136. 10.1007/s002050050037
 3.
Carrillo C, Fife P: Spatial effects in discrete generation population models. J. Math. Biol. 2005, 50(2):161188. 10.1007/s0028500402844
 4.
Chen X: Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differ. Equ. 1997, 2: 125160.
 5.
Coville J, Dáila J, Martinez S: Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differ. Equ. 2008, 244: 30803118. 10.1016/j.jde.2007.11.002
 6.
Fife P: Some nonclassical trends in parabolic and paraboliclike evolutions. In Trends in Nonlinear Analysis. Springer, Berlin; 2003:153191.
 7.
GarcíaMelián J, Rossi JD: On the principal eigenvalue of some nonlocal diffusion problems. J. Differ. Equ. 2009, 246: 2138. 10.1016/j.jde.2008.04.015
 8.
Zhang G, Wang Y: Critical exponent for nonlocal diffusion equations with Dirichlet boundary condition. Math. Comput. Model. 2011, 54: 203209. 10.1016/j.mcm.2011.02.002
 9.
Fang Z, Xu X: Extinction behavior of solutions for the p Laplacian equations with nonlocal sources. Nonlinear Anal., Real World Appl. 2012, 13: 17801789. 10.1016/j.nonrwa.2011.12.008
 10.
Antontsev SN, Shmarev SI: Doubly degenerate parabolic equations with variable nonlinearity II: blowup and extinction in a finite time. Nonlinear Anal. 2014, 95: 483498.
 11.
Gao Y, Gao W: Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity. Bound. Value Probl. 2013., 2013: Article ID 164
 12.
Liu W, Wang M, Wu B: Extinction and decay estimates of solutions for a class of porous medium equations. J. Inequal. Appl. 2007., 2007: Article ID 87650
 13.
Liu W, Wu B: A note on extinction for fast diffusive p Laplacian with sources. Math. Methods Appl. Sci. 2008, 31: 13831386. 10.1002/mma.976
 14.
Mu C, Yan L, Xiao Y: Extinction and nonextinction for the fast diffusion equation. Abstr. Appl. Anal. 2013., 2013: Article ID 747613
 15.
Wu B: Global existence and extinction of weak solutions to a class of semiconductor equations with fast diffusion terms. J. Inequal. Appl. 2008., 2008: Article ID 961045
 16.
Yin J, Li J, Jin C: Nonextinction and critical exponent for a polytropic filtration equation. Nonlinear Anal. 2009, 71: 347357. 10.1016/j.na.2008.10.082
 17.
Liu W: Extinction and nonextinction of solutions for a nonlocal reactiondiffusion problem. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 15
 18.
Chasseigne E, Chaves M, Rossi JD: Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl. 2006, 86: 271291.
 19.
PérezLlanos M, Rossi JD: Blowup for a nonlocal diffusion problem with Neumann boundary conditions and a reaction term. Nonlinear Anal. 2009, 70: 16291640. 10.1016/j.na.2008.02.076
 20.
Andreu F, Mazón JM, Rossi JD, Toledo J Mathematical Surveys and Monographs 165. NonLocal Diffusion Problems 2010.
 21.
Liu W: Extinction properties of solutions for a class of fast diffusive p Laplacian equations. Nonlinear Anal. 2011, 74: 45204532. 10.1016/j.na.2011.04.016
 22.
Andreu F, Mazón JM, Rossi JD, Toledo J: A nonlocal p Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 2009, 40(5):18151851. 10.1137/080720991
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests
Authors’ contributions
JZ carried out critical extinction exponents for a nonlocal reactiondiffusion equation with nonlocal source and interior absorption and drafted the manuscript. BG participated in the design of the study and examined the results carefully. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Gao, B., Zheng, J. Critical extinction exponents for a nonlocal reactiondiffusion equation with nonlocal source and interior absorption. Adv Differ Equ 2014, 19 (2014). https://doi.org/10.1186/16871847201419
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871847201419
Keywords
 nonlocal diffusion
 reactiondiffusion
 extinction
 critical exponent