A wave equation associated with mixed nonhomogeneous conditions: Global existence and asymptotic expansion of solutions

In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution (u,P) of problem (1.1)-(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of α=β=2, Part 2 is devoted the study of the asymptotic behavior of the solution (u,P) as λ1→0+. Finally, in Part 3 we obtain an asymptotic expansion of the solution (u,P) of the problem (1.1)-(1.5) up to order N+12 in four small parameters K, λ, K1, λ1.