Global existence and blow up results for a heat equation with nonlinear nonlocal term

Abstract

We study the nonlocal parabolic equation $$ u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s) \, ds $$ with $p>1$. We assume that $k$ is continuous and there exists $\gamma \in {\mathbb{R}}$ such that $k(\lambda t, \lambda s)=\lambda^{-\gamma}k(t,s)$ for all $\lambda>0$, $0<s<t$. We consider the problem in ${\mathbb{R}}^N$ and a Dirichlet problem in a bounded smooth domain $\Omega$. We analyze the conditions for either blow up or global existence of solutions.