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 \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<\theta \le 2$, $p>1$, $\gamma >0$ and $\mu$ is a nonnegative Radon measure on ${\mathbf{R}}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $\mu$ for the existence of solutions to this problem.

References in corpus (1)

• Well-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation