Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866911769049432064 |
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| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=Δu^m + |x|^σu^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $σ>0$. A new and explicit critical exponent $$ σ_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $σ_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_04500 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs Dynamical Systems We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=Δu^m + |x|^σu^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $σ>0$. A new and explicit critical exponent $$ σ_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $σ_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$. |
| title | Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2103.04500 |