Unconditional uniqueness of Hardy--Hénon parabolic equations on Herz spaces

Abstract

In this paper, we introduce the unconditional uniqueness of solutions in Herz spaces for the Hardy--Hénon 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 ${\dot{K}}_{q,r}^s({\mathbb{R}}^n)$ relaxes the endpoint case $q=\alpha$ and the large interpolation exponent case $r\ge q$ compared to previous results.