Boundary Value Problems
Volume 2009 (2009), Article ID 415709, 16 pages
doi:10.1155/2009/415709

Existence and uniqueness of very singular solution of a degenerate parabolic equation with nonlinear convection

Abstract

We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: (|f′|p−2f′)′+βrf′+αf+(fq)′=0 satisfying a specific decay rate: lim⁡r→∞rα/βf(r)=0 with α:=(p−1)/(pq−2p+2) and β:=(q−p+1)/(pq−2p+2). Here p>2 and q>p−1. Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term ut=(|ux|p−2ux)x+(uq)x defined on the half line [0,+∞).