Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Razvan Gabriel Iagar; Ana Isabel Muñoz; Ariel Sánchez

doi:10.3934/cpaa.2022003

Article Contents

2022, Volume 21, Issue 3: 891-925. Doi: 10.3934/cpaa.2022003

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Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain

^* Corresponding author
^* Corresponding author

Received: August 2021

Early access: December 2021

Published: March 2022

R. I. and A. S. are partially supported by the Spanish project PID2020-115273GB-I00. A. I. M. is partially supported by the Spanish project RTI2018-098743-B-100.

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Abstract

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 \mathbb{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 compactly supported and might present two different types of interface behavior and three different possible 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 $.

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Mathematics Subject Classification: Primary: 35A24, 35B44, 35C06; Secondary: 35K10, 35K57, 35K65.

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Figure 1. Orbits from $ P_0 $ and $ P_2 $ in the phase space for $ \sigma $ small. Experiment for $ m = 3 $, $ p = 0.5 $, $ N = 4 $ and $ \sigma = 3.5 $

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Figure 2. A plot of the regions $ D_1 $, $ D_2 $ and $ D_3 $ in the phase space

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Figure 3. The planes $ (\Pi_1) $ and $ (\Pi_2) $ in the phase space

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Figure 4. Orbits from $ P_2 $ and $ P_0 $ for different values of $ \sigma $. Experiments for $ m = 3 $, $ p = 0.5 $, $ N = 4 $ and $ \sigma = 4.822 $, respectively $ \sigma = 6 $

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