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

TL;DR

This paper investigates the optimal initial data singularities for the Hardy parabolic equation and its fractional variant, establishing conditions under which solutions exist based on the singularity strength at different points.

Contribution

It characterizes the optimal singularity of initial data for solvability of the Hardy parabolic equation, extending results to fractional Laplacian cases and differentiating between singularities at zero and non-zero points.

Findings

Optimal singularity at non-zero points matches Fujita equation.

Weaker singularity needed at zero for solvability.

Results extend to fractional Laplacian cases.

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 $z\neq 0$, 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 $0<\theta<2$.