Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations
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| Format: | Preprint |
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2025
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| author | Chikami, Noboru Ikeda, Masahiro Taniguchi, Koichi Tayachi, Slim |
| author_facet | Chikami, Noboru Ikeda, Masahiro Taniguchi, Koichi Tayachi, Slim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations Chikami, Noboru Ikeda, Masahiro Taniguchi, Koichi Tayachi, Slim Analysis of PDEs Primary 35K05, Secondary 35B40 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. |
| title | Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations |
| topic | Analysis of PDEs Primary 35K05, Secondary 35B40 |
| url | https://arxiv.org/abs/2503.12408 |