Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption

Yulan Wang,Jinsong Hu,Zhaoyin Xiang

doi:10.1155/2012/648067

Abstract

We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.

Highlights

In this paper, we devote our attention to the singularity analysis of the following nonlocal diffusion equation: ut x, tJ x − y u y, t − u x, t dy up x, t − kuq x, t, x ∈ Ω, t > 0, Ω1.1 u x, 0 u0 x, x ∈ Ω.Here Ω is a bounded connected and smooth domain, which contains the origin, and J : RN → R is a nonnegative, bounded, symmetric radially and strictly decreasing function with RN J z dz 1, and p, q, k are all positive constants
From this result we find that the nonlocal diffusion term plays no role when determining the blowup rate and the blowup rate is just same as that of the ODE ut up
For a general domain Ω we can localize the blowup set near any pint in Ω just by taking an initial condition being very large near that point and not so large in the rest of the domain

Summary

Introduction

We devote our attention to the singularity analysis of the following nonlocal diffusion equation: ut x, t. In the Dirichlet boundary case, Xiang et al 13 studied the blowup rate estimates and obtained the following results: if p > max{q, 1} and the solution u x, t of 1.4 blows up at T , there exists constants C > c > 0 such that max u x, τ ≥ c T − t 1/ p−1 ,. For any x0 ∈ Ω and ε > 0, there exists an initial data u0 such that the corresponding solution u x, t of 1.1 blows up at finite time T and its blowup set B u is contained in B x0 {x ∈ Ω; x − x0 < }. When p < 1 or q < 1, the conclusion is validity if u and u are bounded away from 0

Blowup and Global Existence

Blowup Rate Estimate

Blowup Set

Numerical Experiments