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

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 ∂ₜu=Δ uᵐ-|x|σuᵖ, posed for (x,t)N×(0,∞), N≥1, and in the range of exponents 1<m<p<∞, σ>0. We give a complete classification of (singular) self-similar solutions of the form u(x,t)=t-αf(|x|t-β), \ α=σ+2/σ(m-1)+2(p-1), \ β=p-m/σ(m-1)+2(p-1), showing that their form and behavior strongly depends on the critical exponent pF(σ)=m+σ+2/N. For p≥ pF(σ), we prove that all self-similar solutions have a tail as ξ→∞ of one of the forms u(x,t)~ C|x|-(σ+2)/(p-m) or u(x,t)~ (1/p-1)1/(p-1)|x|-σ/(p-1), while for m<p<pF(σ) 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.