A heat equation with a nonlinear nonlocal term in time and singular initial data

Abstract

We consider the general semilinear parabolic equation $u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s)ds$ in $(0,T)\times \mathbb{R}^N$ with $p > 1$. The function $k:\{(t,s); s < t\} \to \mathbb{R}$ is continuous and verifies a scaling property. We prove the existence and non existence results for initial data in the space $L^r(\mathbb{R}^N)$ with $1\leq r < \infty$.