Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions.

In this paper, we consider the following nonlinear Kirchhoff wave equation where $\widetilde{u}_{0}$ , $\widetilde{u}_{1}$ , μ, f, g are given functions and $\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.$ To the 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 particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1) is studied. [ABSTRACT FROM AUTHOR]