Transactions of the American Mathematical Society

Author(s): Alberto Arosio; Stefano Panizzi
Journal: Trans. Amer. Math. Soc. 348 (1996), 305-330.
MSC (1991): Primary 35L70, 35B30; Secondary 34G20
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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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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35L70, 35B30, 34G20

Alberto Arosio
Affiliation: Dipartimento di Matematica, Università di Parma, Parma,Italy

Stefano Panizzi
Affiliation: Dipartimento di Matematica, Università di Parma, Parma,Italy
Email: panizzi@prmat.math.unipr.it

DOI: 10.1090/S0002-9947-96-01532-2
PII: S 0002-9947(96)01532-2
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 {\em Italian Ministero della Università e della Ricerca Scientifica e Tecnologica}.
Copyright of article: Copyright 1996, American Mathematical Society

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