Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.02847 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929408805175296 |
|---|---|
| author | Fujishima, Yohei Ishige, Kazuhiro Kawakami, Tatsuki |
| author_facet | Fujishima, Yohei Ishige, Kazuhiro Kawakami, Tatsuki |
| contents | Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2Δv+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$, and $(μ,ν)$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of solutions for a semilinear parabolic system with singular initial data Fujishima, Yohei Ishige, Kazuhiro Kawakami, Tatsuki Analysis of PDEs Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2Δv+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$, and $(μ,ν)$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces. |
| title | Existence of solutions for a semilinear parabolic system with singular initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.02847 |