Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

Guosheng Zhang,Yifu Wang

doi:10.1155/2013/746086

Abstract

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source , , , , , , and , , which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.

Highlights

Since the long-range effects are taken into account, nonlocal diffusion equations of the form
It is known that (1) shares many properties with the classical heat equation ut = Δu, such that bounded stationary solutions are constant, a maximum principle holds for both of them, and perturbations propagate with infinite speed
In [13, 14], a nonlocal model for diffusion that is analogous to the local porous medium equation is studied

Summary

Introduction

Since the long-range effects are taken into account, nonlocal diffusion equations of the form. There is no regularizing effect in general (see [8]) Another classical equation that has been used to model diffusion Δum with is the well-known porous m > 1. In [13, 14], a nonlocal model for diffusion that is analogous to the local porous medium equation is studied. In this model the probability distribution of jumping from location y to location x is given by J((x − y)/u(y, t))(1/u(y, t)) when u(y, t) > 0 and 0 otherwise. OWrteedasisnum[−e1t,h1a]t, w0 ∈ L1(R) is a nonnegative function In this model, it is assumed that no individual can survive outside of the domain (−L, L). The estimate of the blowup time, the blowup rates, and sets of the solution of the problem (4) are discussed

Existence and Uniqueness

The Proof of Theorem 4

The Proof of Theorem 5