A critical non-homogeneous heat equation with weighted source
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915726896398336 |
|---|---|
| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $σ\ge-2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to\infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $σ=-2$ we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher-KPP equation is derived and employed in order to deduce these properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A critical non-homogeneous heat equation with weighted source Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $σ\ge-2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to\infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $σ=-2$ we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher-KPP equation is derived and employed in order to deduce these properties. |
| title | A critical non-homogeneous heat equation with weighted source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.12902 |