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 ∂tu =Δum−|x|σup, posed for (x, t) ∈ RN × (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 |x| →∞of one of the forms 1/(p−1) u(x,t) ∼ C|x|−(σ+2)/(p−m) or u(x,t) ∼ 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