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

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.