Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.10442 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929339756445696 |
|---|---|
| author | Ageno, Giacomo del Pino, Manuel |
| author_facet | Ageno, Giacomo del Pino, Manuel |
| contents | We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^5,~&\mbox{ in } Ω\times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial Ω\times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^3$. Let $H_γ(x,y)$ be the regular part of the Green function of $-Δ-γ$ in $Ω$, where $γ\in (0,λ_1)$ and $λ_1$ is the first Dirichlet eigenvalue of $-Δ$. Then, given a point $q\in Ω$ such that $3γ(q)<λ_1$, where $$ γ(q)=\sup\{ γ>0: H_γ(q,q)>0 \}, $$ we prove the existence of a non-radial global positive and smooth solution $u(x,t)$ which blows up in infinite time with spike in $q$. The solution has the asymptotic profile $$ u(x,t)\sim 3^{\frac{1}{4}} \bigg(\frac{μ(t)}{μ(t)^2+|x-ξ(t)|^2}\bigg)^{\frac{1}{2}} \quad \text{as}\quad t \to \infty, $$ where $$
-\ln μ(t)= 2γ(q) t(1+o(1)),\quad ξ(t)=q+O\big(μ(t)\big) \quad \text{as}\quad t \to \infty. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10442 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains Ageno, Giacomo del Pino, Manuel Analysis of PDEs 35B40 We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^5,~&\mbox{ in } Ω\times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial Ω\times \mathbb{R}^+,\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^3$. Let $H_γ(x,y)$ be the regular part of the Green function of $-Δ-γ$ in $Ω$, where $γ\in (0,λ_1)$ and $λ_1$ is the first Dirichlet eigenvalue of $-Δ$. Then, given a point $q\in Ω$ such that $3γ(q)<λ_1$, where $$ γ(q)=\sup\{ γ>0: H_γ(q,q)>0 \}, $$ we prove the existence of a non-radial global positive and smooth solution $u(x,t)$ which blows up in infinite time with spike in $q$. The solution has the asymptotic profile $$ u(x,t)\sim 3^{\frac{1}{4}} \bigg(\frac{μ(t)}{μ(t)^2+|x-ξ(t)|^2}\bigg)^{\frac{1}{2}} \quad \text{as}\quad t \to \infty, $$ where $$ -\ln μ(t)= 2γ(q) t(1+o(1)),\quad ξ(t)=q+O\big(μ(t)\big) \quad \text{as}\quad t \to \infty. $$ |
| title | Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains |
| topic | Analysis of PDEs 35B40 |
| url | https://arxiv.org/abs/2301.10442 |