Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations

Jochen  Merker; Aleš Matas

doi:10.3934/proc.2015.0817

Article Contents

2015, Volume 2015: 817-825. Doi: 10.3934/proc.2015.0817 open access

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Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations

Jochen Merker^1, and
Aleš Matas^2,

1.
HTWK Leipzig University of Applied Sciences, Gustav-Freytag-Straße 42a, D - 04277 Leipzig
2.
Katedra matematiky - Západočeská univerzita v Plzni, Univerzitní 22, CZ - 306 14 Plzeň

Received: June 30, 2014

Revised: January 31, 2015

Published: October 2015

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Abstract

The aim of this short note is to study self-similiar radially symmetric solutions of the scalar doubly nonlinear reaction-diffusion equation \begin{equation*} \frac{\partial u^{m-1}}{\partial t} - \Delta_p u = \lambda u^{q-1} \end{equation*} on $\mathbb{R}^n$, where the parameters $1 < m, p,q < \infty$ and $0 < \lambda < \infty$ are fixed. Particularly, for $m < p < q < q_c := p(1+\frac{m-1}{n})$ (where $q_c$ is Fujita's critical exponent of blow-up) we show that there exist self-similar and radially symmetric solutions $u$, which do not blow up in finite time, but instantly become sign-changing for $t>0$ inside some subdomain.

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Mathematics Subject Classification: Primary: 35B05, 35K92; Secondary: 35K65.

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