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 $${\partial }_{t}u=\Delta u^m-|x{|}^{\sigma }{u}^p,$$ posed for $(x,t)\in {\mathbf{R}}^N\times (0,\infty )$, $N\ge 1$, 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 $p\ge p_F(\sigma )$, we prove that all self-similar solutions have a tail as $\xi \to \infty$ of one of the forms $$u(x,t)\sim C|x{|}^{-(\sigma +2)/(p-m)}\mathrm{or}u(x,t)\sim {\left(\frac{1}{p-1}\right)}^{1/(p-1)}|x{|}^{-\sigma /(p-1)},$$ 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.