Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations

TL;DR

The paper addresses the asymptotic behavior of small global solutions to the Hardy-Hénon parabolic equation $\partial_t u - \Delta u = a|x|^{\gamma}|u|^{\alpha-1}u$ by developing a unified local and global theory in weighted Lorentz spaces. It introduces a scale-critical framework with the norm $L^{q,r}_{l}$ and auxiliary spaces $\mathcal{K}^p_k$, enabling fixed-point constructions of mild solutions, global existence for small data, and forward self-similar profiles. The main contributions include a comprehensive classification of asymptotic behaviors for real-valued data (linear vs nonlinear self-similar dominates), a detailed and new complex-valued asymptotic theory featuring nonlinear, linear, and combined Modified Linear regimes, and stability results for the asymptotic profiles. The results unify and extend prior Fujita, Hardy, and Hénon cases across the full spectrum of $\gamma$, providing precise decay rates in weighted Lorentz norms and underpinning the asymptotic self-similarity paradigm with rigorous operator estimates and fixed-point arguments.

Abstract

We construct asymptotically self-similar global solutions to the Hardy-Hénon parabolic equation $\partial_t u - Δu = \pm |x|^γ |u|^{α-1} u$, $α>1$, $γ\in \mathbb{R}$ for a large class of initial data belonging to weighted Lorentz spaces. The solution may be asymptotic to a self-similar solution of the linear heat equation or to a self-similar solution to the Hardy-Hénon parabolic equation depending on the speed of decay of the initial data at infinity. The asymptotic results are new for the Hénon case $γ>0$. We also prove the stability of the asymptotic profiles. Our approach applies for $γ> -\min(2,d)$ and unifies the cases $γ>0$, $γ=0$ and $-\min(2,d)<γ<0$. For complex-valued initial data, a more intricate asymptotic behaviors can be shown; if either one of the real part or the imaginary part of the initial data has a faster spatial decay, then the solution exhibits a combined Nonlinear-"Modified Linear" asymptotic behavior, which is completely new even for the Fujita case $γ=0$. In Appendix, we show the non-existence of local positive solutions for supercritical initial data.

Theorems & Definitions (56)

• Definition 1.1: Mild solution
• Definition 1.2: Auxiliary spaces
• Theorem 1.3: Local well-posedness in the critical space
• Remark 1.4
• Theorem 1.5: Global existence
• Remark 1.6
• Theorem 1.7: Existence of forward self-similar solutions
• Theorem 1.8: Asymptotic stability
• Theorem 1.9: Asymptotically self-similar global solutions
• Remark 1.10