Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation

TL;DR

This paper develops higher order asymptotic expansions for solutions to a nonlinear damped wave equation, providing detailed approximations and estimates under certain initial data conditions.

Contribution

It introduces a method to construct nonlinear approximations of the global solution with respect to data weight, extending previous linear and nonlinear approximation theories.

Findings

Established global solutions with weighted $L^1$ and $L^ Infinity$ estimates.

Derived higher order asymptotic expansions of solutions.

Connected linear solution formulas with nonlinear approximation techniques.

Abstract

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted $L^1$ and $L^\infty$ estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weight of the data. Our proof is based on the approximation formula of the linear solution, which is given in [36], and the nonlinear approximation theory for a nonlinear parabolic equation developed by [18].