Existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions

Abstract

Consider the initial–boundary value problem for the nonlinear wave equation (1) { u t t − ∂ ∂ x ( μ ( x , t ) u x ) + f ( u , u t ) = F ( x , t ) , 0 < x < 1 , 0 < t < T , μ ( 0 , t ) u x ( 0 , t ) = P ( t ) , − μ ( 1 , t ) u x ( 1 , t ) = K 1 | 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 1 , q 1 ≥ 2 , K 1 ≥ 0 are given constants and μ , u 0 , u 1 , f , 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 ≥ 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 , F , g , k ) ∈ H 1 × L 2 × L 1 ( 0 , T ; L 2 ) × L q 0 ′ ( 0 , T ) × L 1 ( 0 , T ) , μ ∈ C 0 ( Q T ¯ ) , μ ( x , t ) ≥ μ 0 > 0 , μ t ∈ L 1 ( 0 , T ; L ∞ ) , μ t ( x , t ) ≤ 0 , a.e. ( x , t ) ∈ Q T ; K 0 , K 1 ≥ 0 ; p 0 , q 0 , p 1 , q 1 ≥ 2 , q 0 ′ = q 0 q 0 − 1 , the function f supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others, we prove that the problem (1) and (2) has a weak solution ( u , P ) . If, in addition, k ∈ W 1 , 1 ( 0 , T ) , p 0 , p 1 ∈ { 2 } ∪ [ 3 , + ∞ ) and some other conditions, then the solution is unique. 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 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 assume ( u 0 , u 1 ) ∈ H 2 × H 1 , f ∈ C 1 ( R 2 ) and some other conditions. Finally, in Part 3, with q 0 = q 1 = 2 ; p 0 , p 1 ≥ N + 1 , f ∈ C N + 1 ( R 2 ) , N ≥ 2 , we obtain an asymptotic expansion of the solution ( u , P ) of the problem (1) and (2) up to order N + 1 in two small parameters K 0 , K 1 .