On the energy decay estimate for the dissipative wave equation with very fast oscillating coefficient and smooth initial data

TL;DR

This paper establishes energy decay estimates for dissipative wave equations with rapidly oscillating coefficients and smooth initial data, using a new condition to handle oscillation effects.

Contribution

It introduces a novel condition for coefficients to effectively evaluate oscillating cancellation, enabling decay estimates for complex wave equations.

Findings

Proved energy decay estimates under new oscillation condition

Handled equations with very fast oscillating coefficients

Applicable to smooth initial data in Gevrey class

Abstract

In this paper we consider energy decay estimates for the Cauchy problems of dissipative wave equations with time dependent coefficients, in particular, the coefficients consisting of weak dissipation and very fast oscillating terms. For such a problem, which have been difficult to deal with in previous research, we prove energy decay estimates by introducing a new condition for the coefficients to evaluate oscillating cancellation of the energy, and smooth initial data such as in the Gevrey class.