Spatial-reiterated Homogenization of Quasi-linear Hyperbolic Equations in a General Deterministic Setting
Gabriel Nguetseng, Hubert Nnang, Nils Svanstedt · Chalmers Publication Library (Chalmers University of Technology) · 2009
Spatial-reiterated deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem. 1. Introduction We study the homogenization (as 0 >>>>: @u @t2 diva x 1 ; x ; t 1 ; Du ( @t ) = f in Q; 0) = 0 in ; @u @t (x; 0) = 0 in ; t) = 0 in @ (0; T ); where is a bounded open set in R (the N -dimensional numerical space of variables x = (x1; ; xN )), T is a positive real number, Q = (0; T ), 1 and are two functions of > 0 such that 1(1) = 2(1) = 1, 1 and are well separated and converge to zero as tends to zero (well separatedness means lim!0 1 = 0 [14]), f is given in L(Q;R), D, div and denote the usual gradient, divergence and Laplacian operators in , and nally, the map a : Ry Rz R R ! R has a p-Laplacian form, 2 p p such that p i(y; z; ) 0 and 0 0. Let > 0. For u 2 Lloc(Q Ry Rz R ), we set (2.1) u(x; t) = u x; t; x 1 ; x ; t 1 (x 2 ; t 2 (0; T )) whenever the right-hand side has meaning. This is the case when u 2 C(Q Ry Rz R ) or u 2 L(Q;A) (1 p 1), where A is a closed vector subspace of B(Ry Rz R ). In the latter case, u belongs to L(Q) with kukLp(Q) kukLp(Q;A) (see [9]). Now, let u 2 C(Q) L(Ry R ;B(Rz )), i.e.,