Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients

TL;DR

This paper investigates the long-term behavior of solutions to a semilinear damped wave equation with variable coefficients, showing they tend to a scaled Gaussian profile under effective damping and perturbation conditions.

Contribution

It introduces a method using scaling variables and energy estimates to analyze asymptotic profiles for the damped wave equation with variable coefficients.

Findings

Solutions approximate a scaled Gaussian profile.

Effective damping ensures the asymptotic behavior.

Perturbations are shown to be negligible asymptotically.

Abstract

We study the asymptotic behavior of solutions for the semilinear damped wave equation with variable coefficients. We prove that if the damping is effective, and the nonlinearity and other lower order terms can be regarded as perturbations, then the solution is approximated by the scaled Gaussian of the corresponding linear parabolic problem. The proof is based on the scaling variables and energy estimates.