Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 67, pp. 1-21.

Title: Radial selfsimilar solutions of a nonlinear 
       Ornstein-Uhlenbeck equation

Authors: Arij Bouzelmate (Faculte des Sciences, Tetouan, Maroc)
         Abdelilah Gmira (Faculte des Sciences, Tetouan, Maroc)
	 Guillermo Reyes (Univ. Carlos III de Madrid, Spain)

Abstract:
 This paper concerns the existence, uniqueness and
 asymptotic properties (as $r=|x|\to\infty$) of radial self-similar
 solutions to the nonlinear Ornstein-Uhlenbeck  equation
 \[
 v_t=\Delta_p  v+x\cdot\nabla (|v|^{q-1}v)
 \]
 in $\mathbb{R}^N\times (0, +\infty)$. Here $q>p-1>1$, $N\geq 1$,
 and $\Delta_p$  denotes the $p$-Laplacian operator.
 These solutions are of the form
 \[
 v(x,t)=t^{-\gamma} U(cxt^{-\sigma}),
 \]
 where $\gamma$ and $\sigma$ 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
 $$
 (|u'|^{p-2}u')'+\frac{N-1}r |u'|^{p-2}u'+\frac{q+1-p}{p} r u'+(q-1)
 r(|u|^{q-1}u)'+u=0
 $$
in $\mathbb{R}_+$. We carry out a careful analysis of this equation and
deduce the corresponding consequences for the Ornstein-Uhlenbeck  equation.
  
Submitted January 11, 2007. Published May 9, 2007.
Math Subject Classifications: 34L30, 35K55, 35K65.
Key Words: p-laplacian; Ornstein-Uhlenbeck diffusion equations;
           self-similar solutions; shooting technique.