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arXiv · 2310.07270

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

Razvan Gabriel Iagar·Philippe Laurençot
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Abstract

Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion equation with strong absorption$$\partial_t u=Δu^m-|x|^σu^q,$$posed for $(t,x)\in(0,\infty)\times\mathbb{R}^N$, with $m>1$, $q\in(0,1)$ and $σ=σ_c:=2(1-q)/(m-1)$ is proved. Looking for radially symmetric solutions of the form$$u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$we show that there exists a unique exponent $β^*\in(0,\infty)$ 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 $σ\in (0,σ_c)$.

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Razvan Gabriel Iagar, Philippe Laurençot. 2023-10-11. Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles. https://arxiv.org/abs/2310.07270

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