Blow up profiles for a reaction-diffusion equation with critical weighted reaction
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arXiv
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| Format: | Preprint |
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2019
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| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^σu^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $σ>0$. In strong contrast with the well-studied equation without the weight (that is $σ=0$), on the one hand we show that for $σ>0$ sufficiently small there exist \emph{multiple self-similar profiles with interface} at a finite point, more precisely, given any positive integer $k$, there exists $δ_k>0$ such that for $σ\in(0,δ_k)$, there are at least $k$ different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for $σ$ sufficiently large, the blow up self-similar profiles with interface \emph{cease to exist}. This unexpected balance between existence of multiple solutions and non-existence of any, when $σ>0$ increases, is due to the effect of the presence of the weight $|x|^σ$, whose influence is the main goal of our study. We also show that for any $σ>0$, there are no blow up profiles supported in the whole space, that is with $u(x,t)>0$ for any $x\in\real$ and $t\in(0,T)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1906_00720 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Blow up profiles for a reaction-diffusion equation with critical weighted reaction Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^σu^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $σ>0$. In strong contrast with the well-studied equation without the weight (that is $σ=0$), on the one hand we show that for $σ>0$ sufficiently small there exist \emph{multiple self-similar profiles with interface} at a finite point, more precisely, given any positive integer $k$, there exists $δ_k>0$ such that for $σ\in(0,δ_k)$, there are at least $k$ different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for $σ$ sufficiently large, the blow up self-similar profiles with interface \emph{cease to exist}. This unexpected balance between existence of multiple solutions and non-existence of any, when $σ>0$ increases, is due to the effect of the presence of the weight $|x|^σ$, whose influence is the main goal of our study. We also show that for any $σ>0$, there are no blow up profiles supported in the whole space, that is with $u(x,t)>0$ for any $x\in\real$ and $t\in(0,T)$. |
| title | Blow up profiles for a reaction-diffusion equation with critical weighted reaction |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1906.00720 |