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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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{align*} u_{tt} - m \left (\int _{_{\partial {\Omega }}} |\nabla _{x}u|^{2} dx \right ) \delta _{x}u= f(x,t) &&& (x\in \Omega , t > 0),\\ u(\cdot ,t)_{|\partial \Omega } =0 &&& (t\geq 0), \end{align*} where $\Omega$ is an open subset of $\mathbb {R}^{n}$ 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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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