Existence and Uniqueness of Very Singular Solution of a Degenerate Parabolic Equation with Nonlinear Convection

We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: (vertical bar f'vertical bar(p-2) f')' + beta rf' + alpha f + (f(q))' = 0 satisfying a specific decay rate: lim(r ->infinity)r(alpha/beta) f(r) = 0 with alpha := (p-1) / (pq-2p+2) and beta := (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 u(t) = (vertical bar u(x)vertical bar(p-2)u(x))(x) + (u(q))(x) defined on the half line [0, +infinity). Copyright (C) 2009 Zhong Bo Fang et al.