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 $(x,t)\in R^N\times (0,\infty )$, $N\ge 1$, 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$u(x,t)=t^{-\alpha }f\left(\right|x\left|t^{-\beta }\right),\alpha =\frac{\sigma +2}{\sigma (m-1)+2(p-1)},\beta =\frac{p-m}{\sigma (m-1)+2(p-1)},$ showing that their form and behavior strongly depends on the critical exponent$p_F\left(\sigma \right)=m+\frac{\sigma +2}{N}.$ For $p\ge p_F\left(\sigma \right)$, we prove that all self-similar solutions have a tail as $\left|x\right|\to \infty$ of one of the forms$u(x,t)\sim C\left|x{|}^{-(\sigma +2)/(p-m)}\mathrm{or}u\right(x,t)\sim {\left(\frac{1}{p-1}\right)}^{1/(p-1)}|x{|}^{-\sigma /(p-1)},$ while for $m<p<p_F\left(\sigma \right)$ we add to the previous the existence and uniqueness of a compactly supported very singular solution. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.