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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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##### References:
 [1] Arosio, A.; Spagnolo, S., Global solutions of the Cauchy problem for a non linear hyperbolic equation, (1982), Universitá di Pisa, Dipartamento di Matematica Roma [2] Aubin, J.P., Un théorème de compacité, C.r. hebd. séanc. acad. sci. Paris, 256, 5042-5044, (1963) · Zbl 0195.13002 [3] Andrade, N.G., On a nonlinear system of partial differential equations, J. math. analysis applic., 91, 119-130, (1983) · Zbl 0523.35026 [4] Ball, J.M., Initial boundary value problems for an extensible beam, J. math. analysis applic., 42, 61-90, (1973) · Zbl 0254.73042 [5] Bernstein, I.N., Sobre uma classe de equaçōes funcionais, Izv. acad. nauk SSSR—math., 4, 17-26, (1960) [6] Brito, E.H., The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Applicable analysis, 13, 219-223, (1982) [7] Dickey, R.W., The initial value problem for a nonlinear semi infinite string, Proc. R. soc. edinb., 82A, 19-26, (1978) · Zbl 0394.45007 [8] Dickey, R.W., Infinite systems of nonlinear oscillation equations with linear damping, SIAM J. appl. math., 19, 459-468, (1970) · Zbl 0218.34015 [9] \scEbihara Y., On the existence of local smooth solutions for some quasilinear hyperbolic equations (to appear). [10] Ebihara, Y., On solutions of semilinear wave equations, Nonlinear analysis, 6., 467-486, (1982) · Zbl 0487.35063 [11] Fitzgibon, W.E., Strongly damped quasilinear evolutions equations, J. math. analysis applic., 79, 536-550, (1981) · Zbl 0476.35040 [12] Lions, J.L., On some questions in boundary value problems of mathematical physics, (1978), Instituto de Matemática, UFRJ Rio de Janeiro, RJ · Zbl 0404.35002 [13] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non lineaires, (1969), Dunod Paris · Zbl 0189.40603 [14] Medeiros, L.A., On a new class of nonlinear wave equations, J. math. analysis applic., 69, 252-262, (1979) · Zbl 0407.35051 [15] \scMedeiros L. A. & \scMilla Miranda M., Local solutions for a nonlinear unilateral problem (to appear). [16] Menzala, G.P., On classical solutions of a quasi linear hyperbolic equation, Nonlinear analysis, 3, 613-629, (1979) [17] Nishida, T., A note on nonlinear vibrations of the elastic string, Mem. fac. engng Kyoto univ., 34, 329-341, (1971) [18] \scNishibara K., On a global solutions of some quasilinear hyperbolic equation (to appear). [19] Pohozaev, S.I., On a class of quasilinear hyperbolic equations, Mat. USSR sbornik, 25, 145-158, (1975) · Zbl 0328.35060 [20] Rivera, P.H., On local strong solutions of a non-linear partial differential equation, Applicable analysis, 10, 93-104, (1980) · Zbl 0451.35042 [21] Ribeiro, N.M., Soluçōes fracas de uma equaçāo da Física matemática, (1979), Instituto de Matemática, UFRJ Rio de Janeiro, RJ
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