LAUSR

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term uα(1−∫Ωuβdx)$u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) $, where Ω$\varOmega$ is a bounded domain in Rn(n≥1)$\mathbb{R}^{n}(n \ge1)$. 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 α$\alpha$, β$\beta$. More precisely, for 1≤α<1+(1−2/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 n≥3$n \ge3$ and β=1$\beta=1$, α<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.