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 $\mathbb{R}^n (n \ge 1)$, with reaction term $u^\alpha(1-m(t))$, where $m(t)$ is the total mass at time $t$. When $\alpha \ge 1$ and the initial mass is greater than or equal to one, the problem has a unique nonnegative classical solution. While if the initial mass is less than one, then the problem admits a unique global solution for $n=1,2$ with any $1 \le \alpha <2$ or $n \ge 3$ with any $1 \le \alpha < 1+2/n$. Moreover, the asymptotic convergence to the solution of the heat equation is proved. Finally, some numerical simulations in dimensions $n=1,2$ are exhibited. Especially, for $\alpha>2$ and the initial mass is less than one, our numerical results show that the solution exists globally in time and the mass tends to one as time goes to infinity.