Deterministic homogenization of weakly damped nonlinear hyperbolic-parabolic equations

Abstract

Abstract. Deterministic homogenization is studied for nonlinear hyperbolic-parabolic equations with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of highly oscillatory nonlinear evolution problems converges to the solution to a homogenized quasilinear hyperbolic-parabolic problem. Mathematics subject classification (1991). 35B40, 46J10.

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

“…Here h ε and g ε are ε × ε r -periodic functions rapidly oscillating. Furthermore, [26,29,30,35] deal with nonlinear wave equations, and in particular, in [26,29], almost periodic settings are studied via Σ-convergence theory developed in [25]. Here (1.7) is called damped wave equations for h ε ≡ 1 and g ε > 0 and it is noteworthy that asymptotic expansions of solutions to damped wave equations are performed with the aid of solutions to diffusion equations (e.g.…”

Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

“…Homogenization of certain nonlinear hyperbolic problems has been performed, e.g., in [4], [26] and [32] having one rapid spatial scale, in [30] with two rapid spatial scales, in [25] having one rapid scale in both space and time, in [22] where two rapid spatial and one rapid temporal scale appear, and in [33] which involves one rapid spatial and two rapid temporal scales. The key homogenization tool in [33], making rapid temporal scales possible to treat, is a compactness result of very weak evolution multiscale convergence type.…”

Reiterated homogenization of hyperbolic-parabolic equations in domains with tiny holes

“…In this direction we quote [4,5,6,7,9,10,12,13,14,22] and references therein. We also mention that Nnang [16] has studied the deterministic homogenization problem for weakly damped nonlinear H-P equations in a fixed domain with ρ = 1.…”