Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition

Abstract

We study the initial-boundary value problem for a nonlinear wave equation given by (1) { u t t − u x x + ∫ 0 t k ( t − s ) u x x ( s ) d s + K | u | p − 2 u + λ | u t | q − 2 u t = f ( x , t ) , 0 < x < 1 , 0 < t < T , u x ( 0 , t ) = u ( 0 , t ) , u x ( 1 , t ) + η u ( 1 , t ) = g ( t ) , u ( x , 0 ) = u ˜ 0 ( x ) , u t ( x , 0 ) = u ˜ 1 ( x ) , where η ≥ 0 ; p ≥ 2 , q ≥ 2 ; K , λ are given constants and u ˜ 0 , u ˜ 1 , f , g , k are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of existence and uniqueness of a weak solution u of problem (1). The proof is based on a Faedo–Galerkin method associated with a priori estimates, weak convergence and compactness techniques. Part 2 is devoted to the study of the asymptotic behavior of the solution u as η → 0 + . Finally, in Part 3 we obtain an asymptotic expansion of the solution u of the problem (1) up to order N + 1 in three small parameters K , λ , η .