On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917968462479360 |
|---|---|
| author | Fino, Ahmad Sobajima, Motohiro |
| author_facet | Fino, Ahmad Sobajima, Motohiro |
| contents | The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-Δu+u^p=0$ in exterior domains $Ω$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the one-dimensional case is formulated with $Ω=(0,\infty)$ which is one of the representative of the connected components in $\mathbb{R}$. One can see that the $C_0$-semigroup for the corresponding linear problem possesses an invariant measure $ϕ(x)\,dx$, where $ϕ$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $ϕ(x)\,dx$ vanishes as $t\to \infty$ if and only if $1<p\leq \min\{2,1+\frac{2}{N}\}$. In the other case $p>\min\{2,1+\frac{2}{N}\}$, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when $N\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains Fino, Ahmad Sobajima, Motohiro Analysis of PDEs 35K20, 35K58, 35B40 The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-Δu+u^p=0$ in exterior domains $Ω$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the one-dimensional case is formulated with $Ω=(0,\infty)$ which is one of the representative of the connected components in $\mathbb{R}$. One can see that the $C_0$-semigroup for the corresponding linear problem possesses an invariant measure $ϕ(x)\,dx$, where $ϕ$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $ϕ(x)\,dx$ vanishes as $t\to \infty$ if and only if $1<p\leq \min\{2,1+\frac{2}{N}\}$. In the other case $p>\min\{2,1+\frac{2}{N}\}$, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when $N\geq 3$. |
| title | On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains |
| topic | Analysis of PDEs 35K20, 35K58, 35B40 |
| url | https://arxiv.org/abs/2503.20480 |