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 $$u_{\\varepsilon }$$ u ε to an effective macroscopic model as $$\\varepsilon \\rightarrow 0$$ ε → 0 . 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.