A nonlocal reaction diffusion equation and its relation with Fujita exponent

Abstract

This paper is concerned with a type of nonlinear reaction-diffusion equation, which arises from the population dynamics. The equation includes a certain type reaction term $u^α(1- σ\int_{\R^n}u^βdx)$ of dimension $n \ge 1$ and $σ>0$. An energy-methods-based proof on the existence of global solutions is presented and the qualitative behavior of solution which is decided by the choice of $α,β$ is exhibited. More precisely, for $1 \le α<1+(1-2/p)β$, where $p$ is the exponent appears in Sobolev's embedding theorem defined in \er{p}, the equation admits a unique global solution for any nonnegative initial data. Especially, in the case of $n\geq 2$ and $β=1$, the exponent $α<1+2/n$ is exactly the well-known Fujita exponent. The global existence result obtained in this paper shows that by switching on the nonlocal effect, i.e., from $σ=0$ to $σ>0$, the solution's behavior differs distinctly, that's, from finite time blow-up to global existence.