Unconditional uniqueness and non-uniqueness for Hardy-H\'enon parabolic equations

TL;DR

This paper investigates the conditions for both uniqueness and non-uniqueness of solutions to Hardy-Hénon parabolic equations, employing weighted Lorentz spaces to extend previous results on related heat equations.

Contribution

It introduces a framework using weighted Lorentz spaces to analyze uniqueness for Hardy-Hénon equations, providing new criteria and extending prior work.

Findings

Established unconditional uniqueness in weighted Lorentz spaces.

Proved non-uniqueness under certain conditions.

Extended results from Lebesgue spaces to weighted Lorentz spaces.

Abstract

We study the problems of uniqueness for Hardy-H\'enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H\'enon type) in the nonlinear term. To deal with the Hardy-H\'enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-H\'enon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.