Global existence and asymptotic behavior of solutions to a nonlocal Fisher–KPP type problem

Abstract

In this work, we consider a nonlocal Fisher–KPP reaction–diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in Rn(n≥1), with reaction term uα(1−m(t)), where m(t) is the total mass at time t. With the help of Pohožaev’s identity, the non-existence of nontrivial stationary solutions with Dirichlet boundary conditions is being shown. When α≥1 and the initial mass is greater than or equal to one, the problem has nonnegative classical solutions. While if the initial mass is less than one, then the problem admits global solutions for n=1,2 with any 1≤α<2 or n≥3 with any 1≤α<1+2/n. Moreover, the asymptotic convergence to the solution of the heat equation is proved.