On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918265295470592 |
|---|---|
| author | Hisa, Kotaro Seki, Yukihiro |
| author_facet | Hisa, Kotaro Seki, Yukihiro |
| contents | We discuss the Hénon parabolic equation $\partial_t u = Δu + |x|^σu^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $σ>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^σ$ vanishes at the origin, solutions seem less likely to blow up at the origin. We construct a solution that blows up at the origin and also carry out an analysis of blow-up rate of solutions. In particular, if $p$ is less than the Joseph--Lundgren exponent, all blow-ups are shown to be of Type I. The lower bound corresponding to Type I rate is also shown for some particular blow-up solutions. As by products, we present a basic result on classification to threshold solutions for every $p>1+σ/N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity Hisa, Kotaro Seki, Yukihiro Analysis of PDEs 35B44 We discuss the Hénon parabolic equation $\partial_t u = Δu + |x|^σu^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $σ>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^σ$ vanishes at the origin, solutions seem less likely to blow up at the origin. We construct a solution that blows up at the origin and also carry out an analysis of blow-up rate of solutions. In particular, if $p$ is less than the Joseph--Lundgren exponent, all blow-ups are shown to be of Type I. The lower bound corresponding to Type I rate is also shown for some particular blow-up solutions. As by products, we present a basic result on classification to threshold solutions for every $p>1+σ/N$. |
| title | On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity |
| topic | Analysis of PDEs 35B44 |
| url | https://arxiv.org/abs/2512.23271 |