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

Abstract

We construct asymptotically self-similar global solutions to the Hardy-Hénon parabolic equation ${\partial }_{t}u-\Delta u=\pm |x{|}^{\gamma }|u{|}^{\alpha -1}u$, $\alpha >1$, $\gamma \in \mathbb{R}$ 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 $\gamma >0$. We also prove the stability of the asymptotic profiles. Our approach applies for $\gamma >-\min (2,d)$ and unifies the cases $\gamma >0$, $\gamma =0$ and $-\min (2,d)<\gamma <0$. 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 $\gamma =0$. In Appendix, we show the non-existence of local positive solutions for supercritical initial data.

61 pages