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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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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