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 uα(1−σ∫Rnuβdx)uα(1−σ∫Rnuβdx) of dimension n≥1n≥1 and σ>0σ>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,βα,β is exhibited. More precisely, for 1≤α<1+(1−2/p)β1≤α<1+(1−2/p)β, where p is the exponent appears in Sobolev's embedding theorem defined in (3), the equation admits global solution for any nonnegative initial data. Especially, in the case of n≥2n≥2 and β=1β=1, the exponent α<1+2/nα<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=0σ=0 to σ>0σ>0, the solution's behavior differs distinctly, that's, from finite time blow-up to global existence.