| Title: |
Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension |
| Authors: |
Iagar, Razvan Gabriel, Muñoz, Ana I., Sánchez, Ariel |
| Publication Year: |
2021 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Analysis of PDEs, Mathematics - Dynamical Systems, 35A24, 35B44, 35C06, 35K10, 35K65 |
| More Details: |
We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $\sigma>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are \emph{compactly supported} and might present two different types of interface behavior and three different possible \emph{good behaviors} near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of $\sigma$. This paper generalizes in dimension $N>1$ previous results by the authors in dimension $N=1$ and also includes some finer classification of the profiles for $\sigma$ large that is new even in dimension $N=1$. |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/2108.09088 |
| Accession Number: |
edsarx.2108.09088 |
| Database: |
arXiv |
