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November  2013, 33(11&12): 5253-5271. doi: 10.3934/dcds.2013.33.5253

An identification problem for a nonlinear one-dimensional wave equation

 1 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano 2 Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 5, 00185 Roma

Received  September 2011 Revised  May 2012 Published  May 2013

We prove the existence of a spatial coefficient in front of a nonlinear term in a one-dimensional wave equation when, in addition to classical initial and boundary condition, an integral mean involving the displacement is prescribed.
Citation: Alfredo Lorenzi, Eugenio Sinestrari. An identification problem for a nonlinear one-dimensional wave equation. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5253-5271. doi: 10.3934/dcds.2013.33.5253
References:
 [1] Aleksander M. Denisov, An inverse problem for a quasilinear wave equation, (Russian) Differ. Uravn., 43 (2007), 1097-1105, 1151-1152; translation in Differ. Equ. 43 (2007), 1123-1131. doi: 10.1134/S0012266107080101.  Google Scholar [2] Ronald Grimmer and Eugenio Sinestrari, Maximum norms in one-dimensional hyperbolic problems, Differential Integral Equations, 5 (1992), 421-432.  Google Scholar [3] Victor Isakov, "Inverse Problems for Partial Differential Equations," Second edition. Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar [4] Aleksey I. Prilepko, Dmitry G. Orlovsky and Igor A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Monographs and Textbooks in Pure and Applied Mathematics, 231, Marcel Dekker, Inc., New York, 2000.  Google Scholar [5] Vladimir G. Romanov, "Inverse Problems of Mathematical Physics," VNU Science Press, b.v., Utrecht, 1987.  Google Scholar [6] Vladimir G. Romanov, "Investigation Methods for Inverse Problems," Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.  Google Scholar [7] Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators, Differential and Integral Equations, 23 (2010), 1-30.  Google Scholar

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References:
 [1] Aleksander M. Denisov, An inverse problem for a quasilinear wave equation, (Russian) Differ. Uravn., 43 (2007), 1097-1105, 1151-1152; translation in Differ. Equ. 43 (2007), 1123-1131. doi: 10.1134/S0012266107080101.  Google Scholar [2] Ronald Grimmer and Eugenio Sinestrari, Maximum norms in one-dimensional hyperbolic problems, Differential Integral Equations, 5 (1992), 421-432.  Google Scholar [3] Victor Isakov, "Inverse Problems for Partial Differential Equations," Second edition. Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar [4] Aleksey I. Prilepko, Dmitry G. Orlovsky and Igor A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Monographs and Textbooks in Pure and Applied Mathematics, 231, Marcel Dekker, Inc., New York, 2000.  Google Scholar [5] Vladimir G. Romanov, "Inverse Problems of Mathematical Physics," VNU Science Press, b.v., Utrecht, 1987.  Google Scholar [6] Vladimir G. Romanov, "Investigation Methods for Inverse Problems," Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.  Google Scholar [7] Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators, Differential and Integral Equations, 23 (2010), 1-30.  Google Scholar
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