Space-time arithmetic quasi-periodic homogenization for damped wave equations

TL;DR

This paper develops a space-time homogenization framework for damped wave equations with oscillating coefficients, revealing how quasi-periodicity and period ratios influence the effective equations and solutions.

Contribution

It introduces a homogenization theorem for quasi-periodic coefficients in damped wave equations, including properties of homogenized matrices and a gradient corrector, using a novel space-time two-scale convergence approach.

Findings

Homogenized equations depend on quasi-periodicity and period ratios.

Cell problems are influenced by the log ratio of periods.

The approach extends homogenization theory to quasi-periodic and log-ratio dependent settings.

Abstract

This paper is concerned with space-time homogenization problems for damped wave equations with spatially periodic oscillating elliptic coefficients and temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main results consist of a homogenization theorem, qualitative properties of homogenized matrices which appear in homogenized equations and a corrector result for gradients of solutions. In particular, homogenized equations and cell problems will turn out to deeply depend on the quasi-periodicity as well as the log ratio of spatial and temporal periods of the coefficients. Even types of equations will change depending on the log ratio and quasi-periodicity. Proofs of the main results are based on a (very weak) space-time two-scale convergence theory.