On the Fisher-KPP Model With Degenerate Diffusion and Nonlocal Nonlinear Sources

Abstract

This paper is devoted to the analysis of non-negative solutions for a generalization of the parabolic equation with porous medium like nonlinear diffusion and nonlinear nonlocal reaction. We investigate under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and aggregation modeled by nonlinear reaction, occurs. Precisely, we exhibit that the qualitative behavior of solutions is decided by the nonlinear diffusion which is chosen in such a way that its scaling and the reaction term coincide, that is, that there is a critical exponent $m+2/n$ for the reaction exponent $\alpha$, solutions exist globally with uniformly upper bounds in the case of (i) for any initial data, (ii) $\alpha >m+2/n$ for small initial data, and (iii) $\alpha =m+2/n$ for small mass capacity $M_0$. In the case of (ii) and (iii), the decay properties of the solution are also obtained. Moreover, numerical simulations are carried out to explore the initial conditions for finite time blow-up of the solutions as well as their blow-up behaviors.