Large time behavior of global solutions of the semilinear damped beam equation with slowly decaying data

Abstract

We discuss the existence of the global solution and its asymptotic profile for the Cauchy problem for the nonlinear damped beam equation. We show that the equation admits a unique global solution with small slowly decaying data e.g. (1+x2)−k2, 0<k≤1. Moreover we show that the solution can be approximated by the solution of the linear heat equation with suitable data. The proof is based upon the estimate for the evolution operator of the linearized equation in the homogeneous Sobolev space.