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Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles

Part of: Stability theory Partial differential equations Parabolic equations and systems Representations of solutions

Published online by Cambridge University Press:  14 March 2024

Razvan Gabriel Iagar Open the ORCID record for Razvan Gabriel Iagar [Opens in a new window]  and
Philippe Laurençot Open the ORCID record for Philippe Laurençot [Opens in a new window]
Razvan Gabriel Iagar
Affiliation:
Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933 Madrid, Spain (razvan.iagar@urjc.es)
Philippe Laurençot
Affiliation:
Laboratoire de Mathématiques (LAMA) UMR 5127, Université Savoie Mont Blanc, CNRS, F–73000 Chambéry, France (philippe.laurencot@univ-smb.fr)
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Abstract

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption

\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]
posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$, with $m>1$, $q\in (0,\,1)$ and $\sigma =\sigma _c:=2(1-q)/ (m-1)$ is proved. Looking for radially symmetric solutions of the form
\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]
we show that there exists a unique exponent $\beta ^*\in (0,\,\infty )$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma \in (0,\,\sigma _c)$.

Keywords

porous medium equationspatially inhomogeneous absorptioneternal solutionsexponential self-similarityglobal solutions

MSC classification

Primary: 35C06: Self-similar solutions
Secondary: 34D05: Asymptotic properties 35A24: Methods of ordinary differential equations 35B33: Critical exponents 35K65: Degenerate parabolic equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 6 , December 2025 , pp. 1998 - 2019
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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