Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 67, pp. 1-22.
Title: Singular solutions of doubly singular
parabolic equations with absorption
Authors: Yuanwei Qi (Hong Kong Univ. of Science & Technology, Hong Kong)
Mingxin Wang (Southeast University, Nanjing, China)
Abstract:
In this paper we study a doubly singular parabolic equation with
absorption,
$$ u_t = \hbox{\rm div} ( |\nabla u^m|^{p-2}\nabla u^m ) -u^q
$$
with $m>0$, $p>1$, $m(p-1)<1$, and $q>1$.
We give a complete classification of solutions, which we call singular,
that are non-negative, non-trivial, continuous in
${\mathbb R}^n \times [0, \infty)\backslash\{(0,0)\} $,
and satisfy $u(x,0)=0$ for all $x\neq 0$.
Applications of similar but simpler equations show that these solutions
are very important in the study of intermediate asymptotic behavior of
general solutions.
Submitted July 15, 2000. Published November 8, 2000.
Math Subject Classifications: 35K65, 35K15.
Key Words: doubly singular parabolic equation; absorption; singular solutions.