A nonlocal reaction diffusion equation and its relation with Fujita exponent

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 $u^\alpha(1- \sigma \int_{\R^n}u^\beta dx)$ of dimension $n \ge 1$ and $\sigma>0$. 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 $\alpha,\beta$ is exhibited. More precisely, for $1 \le \alpha<1+(1-2/p)\beta$, 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 $\beta=1$, the exponent $\alpha<1+2/n$ 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 $\sigma=0$ to $\sigma>0$, the solution's behavior differs distinctly, that's, from finite time blow-up to global existence.