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

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.