A nonlinear wave equation associated with nonlinear boundary conditions: Existence and asymptotic expansion of solutions

Abstract

Consider the initial–boundary value problem for the nonlinear wave equation (1) { u t t − u x x + 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 ) = P ( t ) , − u x ( 1 , t ) = | u ( 1 , t ) | p 1 − 2 u ( 1 , t ) + | u t ( 1 , t ) | q 1 − 2 u t ( 1 , t ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , where p , p 1 , q 1 ≥ 2 , q > 1 , 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 0 | u ( 0 , t ) | p 0 − 2 u ( 0 , t ) + | u t ( 0 , t ) | q 0 − 2 u t ( 0 , t ) − ∫ 0 t k ( t − s ) u ( 0 , s ) d s , where p 0 , q 0 ≥ 2 , K 0 are given constants and g , k are given functions. In this paper, we consider three main parts. In Part 1, under the conditions ( u 0 , u 1 ) ∈ H 1 × L 2 , F ∈ L 2 ( Q T ) , k ∈ W 1 , 1 ( 0 , T ) , g ∈ L q 0 ′ ( 0 , T ) , λ = 1 , K , K 0 ≥ 0 ; p , p 0 , q 0 , p 1 , q 1 ≥ 2 , q > 1 , q 0 ′ = q 0 q 0 − 1 , we prove a theorem of existence and uniqueness of a weak solution ( u , P ) of problem (1) and (2). The proof is based on the Faedo–Galerkin method and the weak compact method associated with a monotone operator. For the case of q 0 = q 1 = 2 , p , q , p 0 , p 1 ≥ 2 , in Part 2 we prove that the unique solution ( u , P ) belongs to ( L ∞ ( 0 , T ; H 2 ) ∩ C 0 ( 0 , T ; H 1 ) ∩ C 1 ( 0 , T ; L 2 ) ) × H 1 ( 0 , T ) , with u t ∈ L ∞ ( 0 , T ; H 1 ) , u t t ∈ L ∞ ( 0 , T ; L 2 ) , u ( 0 , ⋅ ) , u ( 1 , ⋅ ) ∈ H 2 ( 0 , T ) , if we make the assumption that ( u 0 , u 1 ) ∈ H 2 × H 1 and some others. Finally, in Part 3 we obtain an asymptotic expansion of the solution ( u , P ) of the problem (1) and (2) up to order N + 1 in three small parameters K , λ , K 0 .