# American Institute of Mathematical Sciences

June  2013, 2(2): 355-364. doi: 10.3934/eect.2013.2.355

## Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement

 1 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland

Received  October 2012 Revised  February 2013 Published  March 2013

We consider an inverse problem of recovering simultaneously the sound speed and an initial condition for the wave equation from a single Dirichlet data measured on the boundary of the support of the initial condition. The problem is motived from the recently developed hybrid imaging models as well as from classical inverse hyperbolic problems with a single boundary measurement formulation. We establish uniqueness of the recovery and the proof is based on the Carleman estimate and continuous observability inequality for general Riemannian wave equations.
Citation: Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355
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##### References:
 [1] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 [2] Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088 [3] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [4] Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133 [5] Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46 [6] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [7] A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119 [8] Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019179 [9] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [10] Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92 [11] Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469 [12] Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133 [13] Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556 [14] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [15] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [16] Michael Renardy. A backward uniqueness result for the wave equation with absorbing boundary conditions. Evolution Equations & Control Theory, 2015, 4 (3) : 347-353. doi: 10.3934/eect.2015.4.347 [17] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [18] Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control & Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149 [19] Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934/dcdss.2008.1.139 [20] Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071

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