Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

Abstract

This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source $u_t(x,t)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{‍}J((x-y)/u(y,t)\mathrm{})dy-u(x,t)+u^p(x,t)$, $x\in (-L,L)$, $t>\mathrm{0}$, $u(x,t)=\mathrm{0}$, $x\notin (-L,L)$, $t\ge \mathrm{0}$, and $u(x,\mathrm{0})=u_{\mathrm{0}}(x)\ge \mathrm{0}$, $x\in (-L,L)$, which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.