Asymptotic behavior of solutions of a Fisher equation with free boundaries and nonlocal term

Abstract

We study the asymptotic behavior of solutions of a Fisher equation with free boundaries and the nonlocal term (an integral convolution in space). This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We give a dichotomy result, that is, the solution either converges to $1$ locally uniformly in $\mathbb{R}$, or to $0$ uniformly in the occupying domain. Moreover, we give the sharp threshold when the initial data $u_0=\sigma \phi$, that is, there exists $\sigma^*>0$ such that spreading happens when $\sigma>\sigma^*$, and vanishing happens when $\sigma\leq \sigma^*$.

Keywords

• asymptotic behavior of solutions
• free boundary problem
• fisher equation
• nonlocal