Asymptotic profiles for the Cauchy problem of semilinear beam equation with two variable coefficients in the subcritical case

TL;DR

This paper studies the long-term behavior of solutions to a semilinear beam equation with variable coefficients, establishing asymptotic stability of self-similar solutions using energy estimates in weighted spaces.

Contribution

It provides the first proof of asymptotic stability for self-similar solutions in the subcritical case of this semilinear beam equation.

Findings

Proves asymptotic stability of self-similar solutions

Uses energy estimates in weighted spaces in self-similar variables

Analyzes the subcritical case of the semilinear beam equation

Abstract

In this paper, we consider the Cauchy problem for the semilinear beam equation in the subcritical case. We prove an asymptotic stability result of self-similar solutions of the associated parabolic problem. The proof of our results are based in the use of fine energy estimates in weighted spaces rewritten in the parabolic self-similar variables.