Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.11933 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917484873908224 |
|---|---|
| author | Zhang, Kaiqiang Li, Zhiyu |
| author_facet | Zhang, Kaiqiang Li, Zhiyu |
| contents | We consider the semilinear heat equation $$ u_t-Δu=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. (\textit{Ann. Inst. H. Poincaré Anal. Non Linéaire}, \textbf{27} (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2/n$ is the Fujita exponent and $p_S=(n+2)/(n-2)$ (for $n\ge3$) is the critical Sobolev exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11933 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence and nonexistence of global solutions of the semilinear heat equation Zhang, Kaiqiang Li, Zhiyu Analysis of PDEs We consider the semilinear heat equation $$ u_t-Δu=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. (\textit{Ann. Inst. H. Poincaré Anal. Non Linéaire}, \textbf{27} (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2/n$ is the Fujita exponent and $p_S=(n+2)/(n-2)$ (for $n\ge3$) is the critical Sobolev exponent. |
| title | On the existence and nonexistence of global solutions of the semilinear heat equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.11933 |