Singularity and blow-up estimates via Liouville-type theorems for Hardy-H\'enon parabolic equations

TL;DR

This paper derives singularity, decay, and Liouville-type theorems for Hardy-Hénon parabolic equations, providing insights into solution behavior, bounds, and blow-up estimates for both radial and nonradial cases.

Contribution

It introduces new Liouville-type theorems and estimates for Hardy-Hénon parabolic equations, extending understanding of solution singularities and bounds.

Findings

Established space-time singularity and decay estimates.

Proved Liouville-type theorems for solutions.

Derived universal bounds and blow-up estimates.

Abstract

We consider the Hardy-H\'enon parabolic equation $u_t-\Delta u =|x|^a |u|^{p-1}u$ with $p>1$ and $a\in {\mathbb R}$. We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.