Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

Mohamed Jleli,Bessem Samet

doi:10.1155/2015/102639

Abstract

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)),x∈RN,whereJ:RN→R+,p>1, and(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

Highlights

In [1], Garcıa-Melian and Quiros considered the nonlocal diffusion problem: ut (x, t) = (J ∗ u − u) (x, t) + up (x, t), (x, t) ∈ RN × (0, ∞), (1)u (x, 0) = u0 (x), x ∈ RN, where J : RN → R+ is a compactly supported nonnegative function with unit integral, p > 1, and u0 ∈ L1(RN; R+) ∩ L∞(RN; R+)
Equation (1) may model a variety of biological, epidemiological, ecological, and physical phenomena involving media with properties varying in space [2, 3]; similar equations appear, for example, in Ising systems with Glauber dynamics [4]
Yang established that the critical Fujita curve is given by∗

Summary

Introduction

We are first concerned with the following evolution problem: utt (x, t) = (J ∗ u − u) (x, t) + up (x, t) , (x, t) ∈ RN × (0, ∞) , (5) We provide a sufficient condition for the nonexistence of global positive solutions to (5). R+ = [0, ∞), Q = RN × (0, ∞), and J : RN → R+ is a continuous function satisfying the following conditions: (J1) J is symmetric; that is, J(z) = J(−z), for every z ∈ RN.

Results

Conclusion