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

TL;DR

The paper establishes unconditional uniqueness for the Hardy--Hénon parabolic equation with a power-type weight in the nonlinearity within Herz spaces. It develops and employs extended heat-semigroup estimates and Meyer's inequality on $\dot{K}^s_{q,r}$ to control the Duhamel term, and analyzes scale-critical and scale-subcritical regimes via $q_c$ and $Q_c$. Key contributions include (i) a comprehensive extension of linear and nonlinear estimates to Herz spaces, (ii) a Meyers-type inequality in the Herz setting used to handle the nonlinear term, and (iii) a systematic comparison with Lorentz-weighted spaces, showing relaxed endpoint restrictions and broader well-posedness implications. The results provide a robust functional-analytic framework for weighted semilinear heat equations, enabling unconditional uniqueness results under less restrictive endpoint conditions and clarifying the role of Herz spaces in this context.

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 $|x|^γ|u|^{α-1}u$. 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 $q=α$ and the large interpolation exponent case $r\ge q$ compared to previous results.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 2.5
• Proposition 2.6
• Definition 2.7
• Theorem 2.8