Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation

Abstract

This paper concerns the existence, uniqueness and asymptotic properties (as r = |x| → ∞) of radial self-similar solutions to the nonlinear Ornstein-Uhlenbeck equation vt = Δpv + x · ∇(|v|q-1v) in ℝN x (0, +∞). Here q > p - 1 > 1, N ≥ 1, and Δp denotes the p-Laplacian operator. These solutions are of the form v(x, t) = t−γU(cxt-σ), where γ and σ are fixed powers given by the invariance properties of differential equation, while U is a radial function, U(y) = u(r), r = |y|. With the choice c = (q - 1)-1/p, the radial profile u satisfies the nonlinear ordinary differential equation.