On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

Abstract

In this paper, we consider the following nonlinear wave equation (1) { u t t − ∂ ∂ x ( μ ( u ) u x ) = f ( x , t , u , u x , u t ) , 0 < x < 1 , 0 < t < T , u x ( 0 , t ) = g ( t ) , u ( 1 , t ) = 0 , u ( x , 0 ) = u ˜ 0 ( x ) , u t ( x , 0 ) = u ˜ 1 ( x ) , where u ˜ 0 , u ˜ 1 , μ , f , g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of μ ∈ C N + 2 ( R ) , μ 1 ∈ C N + 1 ( R ) , μ ( z ) ≥ μ 0 > 0 , μ 1 ( z ) ≥ 0 , for all z ∈ R , and g ∈ C 3 ( R + ) , f ∈ C N + 1 ( [ 0 , 1 ] × R + × R 3 ) , f 1 ∈ C N ( [ 0 , 1 ] × R + × R 3 ) , a weak solution u ε 1 , ε 2 ( x , t ) having an asymptotic expansion of order N + 1 in two small parameters ε 1 , ε 2 is established for the following equation associated to (1) 2,3: (2) u t t − ∂ ∂ x ( [ μ ( u ) + ε 1 μ 1 ( u ) ] u x ) = f ( x , t , u , u x , u t ) + ε 2 f 1 ( x , t , u , u x , u t ) .