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Local solutions for a nonlinear degenerate hyperbolic equation. (English) Zbl 0594.35068
The author investigates local solutions for the initial-boundary value problem associated to the nonlinear degenerated hyperbolic equation of the type \(u_{tt}-M(\int_{\Omega}| \nabla u|^ 2dx)\Delta u=0,\) which comes from the mathematical description of the vibrations of an elastic stretched string. He proves the existence and uniqueness of solution for the problem in \(L^{\infty}(0,T_ 0;V_{k+2})\) with derivatives \(u'\in L^{\infty}(0,T_ 0;V_{k+2})\) and \(u''\in L^{\infty}(0,T_ 0;V_ k),\) where \(V_ k\) is the domain of the operator \(\Delta^{k/2}\). The main method in the proof is a modified Galerkin approximation.
Reviewer: S.Chen

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L80 Degenerate hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
49M15 Newton-type methods
74H45 Vibrations in dynamical problems in solid mechanics
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