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

Abstract

The paper deals with the initial–boundary value problem for the linear wave equation (1) { u t t − u x x + K u + λ u t = F ( x , t ) , 0 < x < 1 , 0 < t < T , u x ( 0 , t ) = P ( t ) , u ( 1 , t ) = 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , where K , λ are given constants and u 0 , u 1 , F are given functions, and the unknown function u ( x , t ) and the unknown boundary value P ( t ) satisfy the following nonlinear integral equation: (2) P ( t ) = g ( t ) + K 1 | u ( 0 , t ) | α − 2 u ( 0 , t ) + λ 1 | u t ( 0 , t ) | β − 2 u t ( 0 , t ) − ∫ 0 t k ( t − s ) u ( 0 , s ) d s , where K 1 , λ 1 , α , β are given constants and g , k are given functions. 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 λ 1 → 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 + 1 2 in four small parameters K , λ , K 1 , λ 1 .