• Previous Article
    A thermo piezoelectric model: Exponential decay of the total energy
  • DCDS Home
  • This Issue
  • Next Article
    Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace
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 - A, 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. 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. Google Scholar

[3]

Victor Isakov, "Inverse Problems for Partial Differential Equations,", Second edition. Applied Mathematical Sciences, 127 (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 (2000). Google Scholar

[5]

Vladimir G. Romanov, "Inverse Problems of Mathematical Physics,", VNU Science Press, (1987). Google Scholar

[6]

Vladimir G. Romanov, "Investigation Methods for Inverse Problems,", Inverse and Ill-posed Problems Series. VSP, (2002). Google Scholar

[7]

Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators,, Differential and Integral Equations, 23 (2010), 1. Google Scholar

show all references

References:
[1]

Aleksander M. Denisov, An inverse problem for a quasilinear wave equation,, (Russian) Differ. Uravn., 43 (2007), 1097. 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. Google Scholar

[3]

Victor Isakov, "Inverse Problems for Partial Differential Equations,", Second edition. Applied Mathematical Sciences, 127 (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 (2000). Google Scholar

[5]

Vladimir G. Romanov, "Inverse Problems of Mathematical Physics,", VNU Science Press, (1987). Google Scholar

[6]

Vladimir G. Romanov, "Investigation Methods for Inverse Problems,", Inverse and Ill-posed Problems Series. VSP, (2002). Google Scholar

[7]

Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators,, Differential and Integral Equations, 23 (2010), 1. Google Scholar

[1]

Alfredo Lorenzi, Eugenio Sinestrari. Regularity and identification for an integrodifferential one-dimensional hyperbolic equation. Inverse Problems & Imaging, 2009, 3 (3) : 505-536. doi: 10.3934/ipi.2009.3.505

[2]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[3]

François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163

[4]

Franco Obersnel, Pierpaolo Omari. Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 305-320. doi: 10.3934/dcds.2013.33.305

[5]

Annalisa Cesaroni, Matteo Novaga, Andrea Pinamonti. One-dimensional symmetry for semilinear equations with unbounded drift. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2203-2211. doi: 10.3934/cpaa.2013.12.2203

[6]

David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control & Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391

[7]

Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011

[8]

Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040

[9]

Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

[10]

Nikolay A. Gusev. On the one-dimensional continuity equation with a nearly incompressible vector field. Communications on Pure & Applied Analysis, 2019, 18 (2) : 559-568. doi: 10.3934/cpaa.2019028

[11]

Dong-Ho Tsai, Chia-Hsing Nien. On the oscillation behavior of solutions to the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4073-4089. doi: 10.3934/dcds.2019164

[12]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020037

[13]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015

[14]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010

[15]

Denis Mercier, Serge Nicaise. Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 273-300. doi: 10.3934/dcds.1998.4.273

[16]

Inbo Sim. On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition. Communications on Pure & Applied Analysis, 2008, 7 (4) : 905-923. doi: 10.3934/cpaa.2008.7.905

[17]

Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256

[18]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515

[19]

Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649

[20]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]