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On the Well-Posedness of the Kirchhoff String

Authors: Alberto Arosio and Stefano Panizzi
Journal: Trans. Amer. Math. Soc. 348 (1996), 305-330
MSC (1991): Primary 35L70, 35B30; Secondary 34G20
MathSciNet review: 1333386
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Abstract: Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation

\begin{displaymath}% {% \begin{array}{ll} u_{tt}-m \left(\g{ \int_{_{\p{\Omega}}}} |\bigtriangledown_{x}u|^{2} \, dx \right) \bigtriangleup_{x}u= f(x,t) \: & \,(x\in \, \Omega, \, t \G 0),\qquad\qquad\qquad\\ u(\cdot ,t)_{|\partial\Omega} =0 &\,(t\, \geq \,0), \end{array} } \end{displaymath}

where $ \; \Omega \;$ is an open subset of $\; \Reali^{n} \; $ and $\, m \, $ is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.

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  • Am W. F. Ames, Nonlinear partial differential equations in engineering, ch. 3 §10, Academic Press, New York, 1965 MR 35:1235
  • Ar A. Arosio, Averaged evolution equations. The Kirchhoff string and its treatment in scales of Banach spaces, expanded text of a lecture given in ``$2^{\circ}$ workshop on functional-analytic methods in complex analysis '' (Trieste, 1993), W. Tutschke ed., World Scientific, Singapore (to appear)
  • AG A. Arosio and S. Garavaldi, On the mildly degenerate Kirchhoff string, Math. Meth. Appl. Sci. 14 (1991), 177--195 MR 92c:35072
  • AS A. Arosio and S. Spagnolo, Global solutions of the Cauchy problem for a non-linear hyperbolic equation, in ``Nonlinear Partial Differential Equations and their Applications''. Collège de France Seminar, Vol. VI, 1-26, H. Brezis & J.L. Lions eds., Research Notes Math. 109, Pitman, Boston, 1984 MR 86e:35091 598:35062
  • Be S. Bernstein, Sur une classe d'équations fonctionelles aux dérivées partielles (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 17--26 MR 2:102
  • BL N. Bazley & H. Lange, The original Schrödinger equation revisited, Appl. Anal. 21 (1986), 225--233 MR 87m:35062
  • Ca G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157--165; A note on the vibrating string, Quart. Appl. Math. 7 (1949), 97--101 MR 10:458a
  • DS1 P. D'Ancona & S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247-262; On an abstract weakly hyperbolic equation modelling the nonlinear vibrating string, in ``Developments in partial differential equations and applications to mathematical physics'' (Proc.: Ferrara, 1991), G. Buttazzo, G. P. Galdi & L. Zanghirati eds., Plenum Press, 1993
  • DS2 P. D'Ancona & S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions, Arch. Rat. Mech. Anal. 124 (1993), 201--219
  • Di1 R. W. Dickey, Infinite systems of nonlinear oscillations equations related to the string, Proc. Amer. Math. Soc. 23 (1969), 459--468 MR 40:458
  • Di2 R. W. Dickey, The initial value problem for a nonlinear semi-infinite string, Proc. Roy. Soc. Edinburgh, A82 (1-2) (1978), 19--26 MR 80d:45005
  • Eb Y. Ebihara, On the existence of local smooth solutions for some degenerate quasilinear hyperbolic equations, An. Acad. Bras. Ciênc. 57 (1985), 145--152 MR 88c:35105
  • EMM Y. Ebihara, L. A. Medeiros & M. Miranda, Local solution for a nonlinear degenerate hyperbolic equation, Nonlinear Anal. T.M.A. 10 (1986), 27--40 MR 86j:35264
  • Fu D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82--86 MR 35:7170
  • Ga S. Garavaldi, Su un modello integrodifferenziale non lineare della corda/membrana vibrante, Tesi di Laurea, Univ. Parma, December 1989
  • GH J. M. Greenberg & S. C. Hu, The initial-value problem for a stretched string, Quart. Appl. Math. 38 (1980), 289--311 MR 82a:35021
  • K1 T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, (example 5.2) Arch. Rat. Mech. Anal. 58 (1975), 181--205 MR 52:11341
  • K2 T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in ``Spectral Theory and Differential Equations'' (Proc.: Dundee, 1974), 25--70, Lecture Notes Math. 448, Springer, 1975 MR 53:11252; Quasi-linear equations of evolution of hyperbolic type, in ``Hyperbolicity'' (C.I.M.E.: Cortona, 1976), 125--191 (Theorem 3.1, 3.2) Liguori, Napoli, 1977 34S:35052
  • K3 T. Kato, Abstract Differential Equations and Nonlinear Problems (Theorem 5.2) Lezioni Fermiane, Sc. Norm. Sup. Pisa, 1985 MR 88m:34058
  • Ka H. Kauderer, Nichlineare Mechanik, Part 2, B II, §1.88 b, Springer, Berlin, 1958 MR 26:3238
  • Ki G. Kirchhoff, Vorlesungen ober mathematische Physik: Mechanik, ch. 29 §7, Teubner, Leipzig, 1876
  • Li J. L. Lions, On some questions in boundary value problems of mathematical physics, in ``Contemporary developments in continuum mechanics and PDE's'', G.M. de la Penha & L.A. Medeiros eds., North-Holland, Amsterdam, 1978 MR 80a:73003 404:35002
  • LM J. L. Lions and E. Magenes Problèmes aux limites non homogènes et applications Vol. I, ch. 1, 70--77, Dunod, Paris, 1968 MR 40:512
  • Ma M. P. Matos, Mathematical analysis of the nonlinear model for the vibrations of a string, Nonlinear Anal. T.M.A. 17, No.12 (1991), 1125--1137 MR 92k:35191
  • MM L. A. Medeiros & M. Miranda, Solutions for the equations of nonlinear vibrations in Sobolev spaces of fractionary order, Computational Appl. Math. 6 (1987), 257--267 MR 90a:35150
  • Me G. P. Menzala, Une solution d'une équation nonlinéaire d'évolution, C. R. Acad. Sci. Paris 286 (1978), 273--275; On a classical solutions of quasilinear hyperbolic equation, Nonlinear Anal. T.M.A. 3 (1979), 613--627 MR 81e:35075
  • Na R. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vibration 8 (1968), 134--146 164:267
  • NM A. H. Nayfeh & D. T. Mook, Nonlinear oscillations, §7.4 - §7.6, Wiley-Interscience, New York, 1979 MR 80m:70002
  • Ni T. Nishida, Nonlinear vibrations of an elastic string II, unpublished manuscript (1971--1978)
  • Nis K. Nishihara, On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math. 7 (1984), 437--459 MR 86g:35124
  • Op D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoustic Soc. Amer. 32 (1960), 1529--1538 MR 22:11649
  • Po1 S. I. Pokhozhaev, On a class of quasilinear hyperbolic equations, Mat. Sbornik 96 (138) (1) (1975), 152--166) (English transl.: Math. USSR Sbornik 25 (1975), 145--158) MR 51:6167 309:35051
  • Po2 S. I. Pokhozhaev, The Kirchhoff quasilinear hyperbolic equation, Differentsial'nye Uravneniya 21, No.1 (1985), 101--108 (English transl.: Differential Equations 21 (1985), 82--87) MR 86h:35056
  • Ri P. H. Rivera Rodriguez, On a nonlinear hyperbolic equation, An. Acad. Brasil. Ciênc. 50 (1978), 133--135 MR 58:17399; On local strong solutions of a nonlinear partial differential equation, Appl. Anal. 10 (1980), 93--104 MR 81h:34070
  • Ru W. Rudin, Functional Analysis, ch. 13, Mc Graw-Hill, New York, 1973 MR 51:1315
  • Ya Y. Yamada, Some nonlinear degenerate wave equations, Nonlinear Anal. T.M.A. 10 (1987), 1155--1168 641:35044 MR 89a:35142
  • Yam T. Yamazaki, On local solutions of some quasilinear degenerate hyperbolic equations, Funkcial. Ekvac. 31 (1988), 439--457 MR 90g:35109

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Additional Information

Stefano Panizzi

Keywords: Well-posedness, quasilinear hyperbolic equation, extensible string, local existence, Ritz-Galerkin approximation
Received by editor(s): April 25, 1994
Received by editor(s) in revised form: January 30, 1995
Additional Notes: The research was supported by the 40% funds of the Italian Ministero della Università e della Ricerca Scientifica e Tecnologica.
Article copyright: © Copyright 1996 American Mathematical Society

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