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 partial derivative(t)u = Delta(m )(u)- |x|(sigma)u(p), posed for (x, t) is an element of R(N )x (0,infinity), N >= 1, and in the range of exponents 1 < m < p < infinity, 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 = sigma+2/sigma(m-1) + 2(p-1), beta = p-m/sigma(m-1) + 2(p-1), showing that their form and behavior strongly depends on the critical exponent p(F)(sigma) = m + sigma+2/N. For p >= p(F)(sigma), we prove that all self-similar solutions have a tail as |x| -> infinity of one of the forms u(x, t) similar to C|x|(-(sigma+2)/(p-m) )or u(x,t)similar to(1/p-1)(1/(p-1))|x|(-sigma/(p-1)), while for m < p < p(F)(sigma) 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by-nc-nd /4 .0/).