Extinction and non-extinction of solutions for a nonlocal reaction-diffusion problem

We investigate extinction properties of solutions for the homogeneous Dirichlet bound- ary value problem of the nonlocal reaction-diffusion equationut−du+ku p = R u q (x,t)dx with p,q ∈ (0,1) and k,d > 0. We show that q = p is the critical extinction exponent. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived.


Introduction and main results
This paper is devoted to the extinction properties of solutions for the following diffusion equation with nonlocal reaction u t − d∆u + ku p = Ω u q (x, t) dx, x ∈ Ω, t > 0, (1.1) subject to the initial and boundary value conditions u(x, t) = 0, x ∈ ∂Ω, t > 0, ( u(x, 0) = u 0 (x), x ∈ Ω, ( where p, q ∈ (0, 1), k, d > 0, Ω ⊂ R N (N > 2) is an bounded domain with smooth boundary and u 0 (x) ∈ L ∞ (Ω) ∩ W 1,2 0 (Ω) is a nonzero non-negative function.Many physical phenomena were formulated into nonlocal mathematical models ( [2,3,6,7]) and there are a large number of papers dealing with the reaction-diffusion equations with nonlocal reactions or nonlocal boundary conditions (see [18,20,21,23] and the references therein).In particular, M. Wang and Y. Wang [23] studied problem (1.1)-(1.3)for p, q ∈ [1, +∞) and concluded that: the blow-up occurs for large initial data if q > p ≥ 1 while all solutions exist globally if 1 ≤ q < p; in case of p = q, the issue depends on the comparison of |Ω| and k.For further studies of problem (1.1)- (1.3) we refer the read to [1,13,14,19,26] and the references therein.In all the above works, p, q ∈ [1, +∞) was assumed.EJQTDE, 2010 No. 15, p. 1 Extinction is the phenomenon whereby the evolution of some nontrivial initial data u 0 (x) produces a nontrivial solution u(x, t) in a time interval 0 < t < T and then u(x, t) ≡ 0 for all (x, t) ∈ Ω × [T, +∞).It is an important property of solutions for many evolution equations which have been studied extensively by many researchers.Especially, there are some papers concerning the extinction for the following semilinear parabolic equation for special cases where p ∈ (0, 1) and q ∈ (0, 1].In case λ = 0, it is well-known that solutions of problem (1.2)-(1.4)vanishes within a finite time.Evans and Knerr [9] established this for the Cauchy problem by constructing a suitable comparison function.Fukuda [10] studied problem (1.2)- (1.4) with λ > 0 and q = 1 and concluded that: when λ < λ 1 , the term ∆u dominates the term λu so that solutions of problem (1.2)-(1.4)behave the same as those of (1.2)-(1.4)with ) grow up to infinity as t → ∞.Here, λ 1 is the first eigenvalue of −∆ with zero Dirichlet boundary condition and φ(x) > 0 in Ω with max x∈Ω φ(x) = 1 is the eigenfunction corresponding to the eigenvalue λ 1 .Yan and Mu [24] investigated problem (1.2)-(1.4)with 0 < p < q < 1 and N > 2(q − p)/(1 − p) and obtained that the non-negative weak solution of problem (1.2)- (1.4) vanishes in finite time for any initial data provided that k is appropriately large.For papers concerning the extinction for the porous medium equation or the p-Laplacian equation, we refer the reader to [8,11,12,15,16,22,25] and the references therein.Recently, the present author [17] considered the extinction properties of solutions for the homogeneous Dirichlet boundary value problem of the p-Laplacian equation u t − div |∇u| p−2 ∇u + βu q = λu r , x ∈ Ω, t > 0.
But as far as we know, no work is found to deal with the extinction properties of solutions for problem (1.1)-(1.3)which contains a nonlocal reaction term.
The purpose of the present paper is to investigate the extinction properties of solutions for the nonlocal reaction-diffusion problem (1.1)-(1.3).Our results below show that q = p is the critical extinction exponent for the weak solution of problem (1.1)-(1.3):if 0 < p < q < 1, the non-negative weak solution vanishes in finite time provided that |Ω| is appropriately small or k is appropriately large; if 0 < q < p < 1, the weak solution cannot vanish in finite time for any non-negative initial data; if 0 < q = p < 1, the weak solution cannot vanish in finite time for any non-negative initial data when k < Ω ψ q (x)dx/M q (≤ |Ω|), while it vanishes in finite time for any initial data u 0 when k > |Ω|.Here ψ(x) is the unique positive solution of the linear elliptic problem and M = max x∈Ω ψ(x).This is quite different from that of local reaction case, in which the first eigenvalue of the Dirichlet problem plays a role in the critical case (see [8,12,15,22,25]).
EJQTDE, 2010 No. 15, p. 2 Moreover, the precise decay estimates of solutions before the occurrence of the extinction will be derived.
We now state our main results.
vanishes in finite time provided that the initial data u 0 (or |Ω|) is appropriately small or k is appropriately large. 2 vanishes in finite time for any initial data provided that |Ω| is appropriately small or k is appropriately large. 3 vanishes in finite time for any initial data provided that |Ω| is appropriately small or k is appropriately large.
Moreover, one has ) are positive constants to be given in the proof, α 1 > d 1 λ 1 and Remark 1 One can see from the proof below that the restriction N > 4(q in the case 3) can be extended to N > 2(q − p)/(1 − p).This has been proved in [24] for the local reaction case.
1) If k > |Ω|, the non-negative weak solution of problem (1.1)-(1.3)vanishes in finite time for any initial data u 0 .Moreover, one has where d 4 and k 4 are positive constants to be given in the proof.
3) If k = Ω ψ q (x)dx/M q , then the weak solution of problem (1.1)-(1.3)cannot vanish in finite time for any identically positive initial data.
Theorem 3 Assume that 0 < q < p < 1, then the weak solution of (1.1)-(1.3)cannot vanish in finite time for any non-negative initial data.
Remark 2 One can conclude from Theorems 1-3 that q = p is the critical extinction exponent of solutions for problem (1.1)-(1.3).
The rest of the paper is organized as follows.In Section 2, we will give some preliminary lemmas.We will prove Theorems 1-3 in Section 3-5.

Preliminary
Let • p and • 1,p denote L p (Ω) and W 1,p (Ω) norms respectively, 1 ≤ p ≤ ∞.Before proving our main results, we will give some preliminary lemmas which are of crucial importance in the proofs.We first give the following comparison principle, which can be proved as in [22,23,25].
The following inequality problem is often used to derive extinction of solutions (see [22,25]).
where α > 0 is a constant and k ∈ (0, 1).Due to the nature of our problem, we would like to use the following lemmas which are of crucial importance in the proofs of decay estimates.
Proof.It is easy to prove that the following algebraic equation αy k + βy = γ y m has unique positive solution (denoted by y * ).
We first consider the case y 0 > 0. By considering the sign of y ′ (t) via y(t) at [0, y * ), we see that: if 0 < y 0 < y * , then y(t) is increasing with respect to t > 0; if y 0 > y * , then y(t) is decreasing with respect to t > 0. Therefore, solution with non-negative initial value y 0 remains positive and of course approaches y * as t → +∞.
We now only need to prove the above mentioned claim.Indeed, if it is not true, then 0 < z(t) < ε for all t ∈ (−∞, 0).Since 0 < m < k < 1 and z ′ (t) > 0 for all t ∈ R, there is a Integrating the above inequality on (t, t 1 ), we get which causes a contradiction as t → −∞.
3) For k = Ω ψ q (x)dx/M q , let w(x, t) = h(t)ψ(x), where h(t) satisfies the ODE problem Then, for any identically positive initial data, we can choose h 0 sufficiently small such that h 0 ψ(x) ≤ u 0 (x).According to Lemma 1, we get that w(x, t) is a non-extinction subsolution of problem (1.1)-(1.3).