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The Galerkin-averaging method for nonlinear, undamped continuous systems. (English) Zbl 0638.35057

This paper is concerned with initial-boundary value problems for nonlinear evolution equations of the form: \[ (1)\quad u_{tt}+Au=\epsilon f(u,u_ t,t),\quad t\geq 0. \] For these problems, the authors set up a first and second order approximation theory, by means of Galerkin approximation and averaging procedures. They, also, study the difference between two cases: the case with nontrivial resonance at first order, illustrated by the nonlinear wave equation \(u_{tt}-u_{xx}=\epsilon u^ 3\), and the case with no resonance at first order, illustrated by the Klein-Gordon equation \(u_{tt}- u_{xx}+u=\epsilon u^ 3.\)
Reviewer: D.Huet

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A35 Theoretical approximation in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:

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