ASYMPTOTIC ANALYSIS OF A NONLINEAR PARTIAL DIFFERENTIAL EQUATION IN A SEMICYLINDER

Abstract. Small solutions of a nonlinear partial differential equation in a semi-infinite cylinder will be studied. We consider the asymptotic behaviour of these solutions at infinity under Neumann boundary condition as well as Dirichlet boundary condition. In the Neumann case it can be shown that any solution small enough either vanishes at infinity or tends to a nonzero periodic solution of a nonlinear ordinary differential equation. In the Dirichlet case every solution small enough vanishes. 1. Introduction. Let Ω be a bounded domain in Rn−1 with C2-boundary. We define the semi-infinite cylinder C+ = {x = (x′, xn) : x ′ ∈ Ω, xn> 0}. In Section 2 we consider bounded solutions of the equation ∆U + q(U)U = H in C+ (1.1