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Pré-Publication, Document De Travail Année : 2023

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

Razvan Gabriel Iagar
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Philippe Laurençot

Résumé

Existence of specific \emph{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 $(t,x)\in(0,\infty)\times\real^N$, 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 $$ u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}), \qquad \alpha=\frac{2}{m-1}\beta, $$ 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 $\sigma\in (0,\sigma_c)$.
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hal-04235311 , version 1 (10-10-2023)

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Razvan Gabriel Iagar, Philippe Laurençot. Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles. 2023. ⟨hal-04235311⟩
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