Homogenization of Nonlinear Dissipative Wave Equations with Multiscale Temporal Oscillations

Abstract

We develop a homogenization theory for nonlinear hyperbolic equations with rapid oscillations in space and multiple microscopic time scales. The model features a semilinear wave equation with oscillatory coefficients and separated nonlinearities: an amplitude-dependent potential and a gradient-dependent forcing term. Under appropriate periodicity and structure conditions, we prove convergence of solutions uε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\varepsilon }$$\end{document} to an effective macroscopic model as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}. The limit problem consists of a homogenized wave operator with a nonlocal-in-time dissipation operator arising from the fast temporal oscillations. Our analysis uses multiscale convergence techniques adapted for wave propagation with multiple temporal scales, extending homogenization theory to a class of nonlinear hyperbolic problems not previously covered.