Optimal singularities of initial data for solvability of the Hardy parabolic equation

Abstract

We consider the Cauchy problem for the Hardy parabolic equation ${\partial }_{t}u-\Delta u=|x{|}^{-\gamma }{u}^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\ne 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation ${\partial }_{t}u-\Delta u=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case ${\partial }_{t}u+(-\Delta {)}^{\theta /2}u=|x{|}^{-\gamma }{u}^p$ with $0<\theta <2$.