• 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 and 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.

[2]

Ronald Grimmer and Eugenio Sinestrari, Maximum norms in one-dimensional hyperbolic problems, Differential Integral Equations, 5 (1992), 421-432.

[3]

Victor Isakov, "Inverse Problems for Partial Differential Equations," Second edition. Applied Mathematical Sciences, 127, Springer, New York, 2006.

[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.

[5]

Vladimir G. Romanov, "Inverse Problems of Mathematical Physics," VNU Science Press, b.v., Utrecht, 1987.

[6]

Vladimir G. Romanov, "Investigation Methods for Inverse Problems," Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.

[7]

Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators, Differential and Integral Equations, 23 (2010), 1-30.

show all references

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.

[2]

Ronald Grimmer and Eugenio Sinestrari, Maximum norms in one-dimensional hyperbolic problems, Differential Integral Equations, 5 (1992), 421-432.

[3]

Victor Isakov, "Inverse Problems for Partial Differential Equations," Second edition. Applied Mathematical Sciences, 127, Springer, New York, 2006.

[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.

[5]

Vladimir G. Romanov, "Inverse Problems of Mathematical Physics," VNU Science Press, b.v., Utrecht, 1987.

[6]

Vladimir G. Romanov, "Investigation Methods for Inverse Problems," Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002.

[7]

Eugenio Sinestrari, Semilinear differential and integrodifferential equations with Hille-Yosida operators, Differential and Integral Equations, 23 (2010), 1-30.

[1]

Alfredo Lorenzi, Eugenio Sinestrari. Regularity and identification for an integrodifferential one-dimensional hyperbolic equation. Inverse Problems and 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 and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[3]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[4]

Zhi-Xue Zhao, Mapundi K. Banda, Bao-Zhu Guo. Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2533-2554. doi: 10.3934/dcdsb.2020021

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[12]

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

[13]

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

[14]

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

[15]

Umberto Biccari, Mahamadi Warma, Enrique Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1949-1978. doi: 10.3934/cpaa.2020086

[16]

Lili Fan, Lizhi Ruan, Wei Xiang. Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1971-1999. doi: 10.3934/dcds.2020349

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]