Homogenization and uniform stabilization for a nonlinear hyperbolic equation in domains with holes of small capacity. (English) Zbl 1053.35021

Summary: We study the homogenization and uniform decay of the nonlinear hyperbolic equation \[ \partial_{tt}u_\varepsilon-\Delta u_\varepsilon+ F(x,t,\partial_t u_\varepsilon,\nabla u_\varepsilon)=0\quad \text{in } \Omega_\varepsilon \times (0,+\infty) \] where \(\Omega_\varepsilon\) is a domain containing holes with small capacity (i.e., the holes are smaller than a critical size). The homogenizations proofs are based on the abstract framework introduced by D. Cioranescu and F. Murat [Nonlinear partial differential equations and their applications, Coll. de France Semin., Vol. III. Res. Notes Math. 70, 154–178 (1982; Zbl 0498.35034)] for the study of homogenization of elliptic problems. Moreover, uniform decay rates are obtained by considering the perturbed energy method developed by A. Haraux and E. Zuazua [Arch. Ration. Mech. Anal. 100, No. 2, 191–206 (1988; Zbl 0654.35070)].


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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