Unconditional uniqueness of Hardy--H\'enon parabolic equations on Herz spaces

TL;DR

This paper establishes the unconditional uniqueness of solutions to the Hardy--Hénon parabolic equation within Herz spaces, extending previous results by relaxing endpoint and interpolation conditions.

Contribution

It introduces the unconditional uniqueness of solutions in Herz spaces for the Hardy--Hénon parabolic equation, handling the power-type weight in the nonlinear term more effectively.

Findings

Unconditional uniqueness of solutions in Herz spaces.

Relaxation of endpoint case $q=\alpha$ and large interpolation exponent case $r\ge q$.

Extension of previous results to broader Herz space settings.

Abstract

In this paper, we introduce the unconditional uniqueness of solutions in Herz spaces for the Hardy--H\'enon parabolic equation, which is a semilinear heat equation with a power-type weight in the nonlinear term $|x|^\gamma|u|^{\alpha-1}u$. It is expected that the power-type weight in the nonlinear term can be effectively handled within Herz spaces. In fact, our result in Herz spaces $\dot{K}^s_{q,r}({\mathbb R}^n)$ relaxes the endpoint case $q=\alpha$ and the large interpolation exponent case $r\ge q$ compared to previous results.