Electron. J. Diff. Eqns., Vol. 2007(2007), No. 67, pp. 1-21.

Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation

Arij Bouzelmate, Abdelilah Gmira, Guillermo Reyes

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 greater than p-1 greater than 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.

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Arij Bouzelmate
Département de Mathématiques et Informatique
Faculté des Sciences, BP 2121, Tétouan, Maroc
email: bouzelmatearij@yahoo.fr
Abdelilah Gmira
Département de Mathématiques et Informatique
Faculté des Sciences, BP 2121, Tétouan, Maroc
email: gmira@fst.ac.ma or gmira.i@menara.ma
  Guillermo Reyes
Departamento de Matemáticas
Universidad Carlos III de Madrid, Leganés, Madrid 28911, Spain
email: greyes@math.uc3m.es

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