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

Abstract

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 $(t,x)\in (0,\infty )\times {\mathbb{R}}^N$, 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$$u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}),\alpha =\frac{2}{m-1}\beta ,$$we show that there exists a unique exponent ${\beta }^{\ast }\in (0,\infty )$ 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 $f_A^{\prime }(0)=0$. 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 $\sigma \in (0,{\sigma }_c)$.