Global Solutions to a Nonlocal Fisher-KPP Type Problem

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\). 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 \(\alpha\), \(\beta\). More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\), \(\alpha<1+2/n\) 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.