Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations

Abstract

We study the asymptotic behaviour of the solutions of the parabolic equation (1) ∂u/∂t−Lu+a(x)|u|^q−1u=0 or the elliptic equation (2) ∂²u/∂t²+Lu−a(x)|u|^q−1u=0 in Ω×(0,∞) when Ω is bounded, u satisfies the Neumann boundary condition in ∂Ω×(0,∞),L is a linear strongly elliptic operator in Ω,q is bigger than 1 and a(x)≥0. We also study the vanishing property of t $\mapsto $ u(x,t) when 0 < q < 1.