On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profiles

Abstract

We study the initial value problem for some semilinear damped beam equation. In Takeda and Yoshikawa (submitted for publication) [9] unique global existence of a decaying solution for the problem and the smoothing effect of the solution was shown. In this article we shall give the asymptotic profiles of the solution. As a result, we observe that the decay estimates are optimal. Moreover, considering the higher order expansion of the solution, we observe more detailed information such as the contribution of the nonlinear term to the solution as t→∞.