A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions

Abstract

This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbf{R}^N\times(0,\infty)$, $N\geq1$, and in the range of exponents $1<m<p<\infty$, $\sigma>0$. We give a complete classification of (singular) self-similar solutions of the form $$ u(x,t)=t^{-\alpha}f(|x|t^{-\beta}), \ \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \ \beta=\frac{p-m}{\sigma(m-1)+2(p-1)}, $$ showing that their form and behavior strongly depends on the critical exponent $$ p_F(\sigma)=m+\frac{\sigma+2}{N}. $$ For $p\geq p_F(\sigma)$, we prove that all self-similar solutions have a tail as $\xi\to\infty$ of one of the forms $$ u(x,t)\sim C|x|^{-(\sigma+2)/(p-m)} \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-\sigma/(p-1)}, $$ while for $m<p<p_F(\sigma)$ we add to the previous the \emph{existence and uniqueness} of a \emph{compactly supported very singular solution}. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.