Global Solutions to a Nonlocal Fisher-KPP Type Problem

Abstract

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term uź(1źźΩuβdx)$u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) $, where Ω$\varOmega$ is a bounded domain in Rn(nź1)$\mathbb{R}^{n}(n \ge1)$. The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of ź$\alpha$, β$\beta$. More precisely, for 1≤ź<1+(1ź2/p)β$1 \le\alpha <1+ ( 1-2/p ) \beta$, where p$p$ is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of nź3$n \ge3$ and β=1$\beta=1$, ź<1+2/n$\alpha<1+2/n$ is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109---124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109---124, 1966), this paper will give an opposite result to our nonlocal problem.