Existence and nonexistence of solutions to the Hardy parabolic equation

Abstract

In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=\mu \quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<\theta\le2$, $p>1$, $\gamma>0$ and $\mu$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $\mu$ for the existence of solutions to this problem.