Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866910314265575424 |
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| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ in the range of exponents $1<p<m$ and $σ>0$. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient $|x|^σ$ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for $σ\sim0$, blow up profiles have similar behavior to the well-established profiles for the homogeneous case $σ=0$, and typically \emph{global blow up} occurs, while for $σ>0$ sufficiently large, there exist blow up profiles for which blow up \emph{occurs only at space infinity}, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1811_10330 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ in the range of exponents $1<p<m$ and $σ>0$. We classify blow up solutions in self-similar form, that are likely to represent typical blow up patterns for general solutions. We thus show that the non-homogeneous coefficient $|x|^σ$ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for $σ\sim0$, blow up profiles have similar behavior to the well-established profiles for the homogeneous case $σ=0$, and typically \emph{global blow up} occurs, while for $σ>0$ sufficiently large, there exist blow up profiles for which blow up \emph{occurs only at space infinity}, in strong contrast with the homogeneous case. This work is a part of a larger program of understanding the influence of unbounded weights on the blow up behavior for reaction-diffusion equations. |
| title | Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1811.10330 |