In this paper we consider a nonlinear parabolic equation of the following type:(P)      ∂u/∂t - div(|∇p|p-2 ∇u) = h(x,u)with Dirichlet boundary conditions and initial data in the case when 1 < p < 2.We construct supersolutions of (P), and by use of them, we prove that for tn → +∞, the solution of (P) converges to some solution of the elliptic equation associated with (P).

On considère le problème de Dirichlet :$$(d{d}^{c}u{)}^n=0\mathrm{dans}B$$et$$u{|}_{\partial B}=\varphi$$où $B$ désigne la boule unité de ${ℂ}^n.$Nous donnons une démonstration simple du fait que si $\varphi \in C^{1,1}\left(\partial B\right)$, alors $u\in C^{1,1}\left(B\right)$; de plus la croissance du coefficient de Lipschitz de la différentielle de $u$ est contrôlée par l’inverse de la distance au bord.

We study uniformly elliptic fully nonlinear equations $F(D^{2}u,Du,u,x)=0$, and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.