Singular Solutions of Doubly Singular Parabolic Equations with Absorption

Abstract

In this paper we study a doubly singular parabolic equation with absorption, u(t) = div(|∇um |p-2∇um) - uq 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 ℝⁿ x [0,∞) \ {(0,0)}, and satisfy u(x,0) = 0 for all x ≠ 0. Applications of similar but simpler equations show that these solutions are very important in the study of intermediate asymptotic behavior of general solutions.