Title

Classification Of Singular Solutions Of Porous Media Equations With Absorption

Abstract

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption ut = Δum - uq in ℝn × (0, ∞). We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝn × (0, ∞)\{(0,0)}, and satisfy u(x,0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫ ℝnu(.,t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫ℝn u∞(.,t) → ∞ as ↘ 0. In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u (c2) when c1 < c2, and u(c) ↗ u∞ as c↗ ∞. Furthermore, u ∞ is self-similar in the sense that u∞(x,t) = t-αw(|x|t-αβ) for α = 1/(q - 1), β= 1/2(q - m), and some smooth function w defined on [0, ∞), so that ∫ℝn unβ∞(.,t) is a finite positive constant independent of t > 0. © 2005 The Royal Society of Edinburgh.

Publication Date

1-1-2005

Publication Title

Royal Society of Edinburgh - Proceedings A

Volume

135

Issue

3

Number of Pages

563-584

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1017/S0308210500004005

Socpus ID

22544436685 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/22544436685

This document is currently not available here.

Share

COinS