# American Institute of Mathematical Sciences

June  2018, 12(3): 745-772. doi: 10.3934/ipi.2018032

## Determination of singular time-dependent coefficients for wave equations from full and partial data

 1 Beijing Computational Science Research Center, Building 9, East Zone, ZPark Ⅱ, No.10 Xibeiwang East Road, Haidian District, Beijing 100193, China 2 Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

* Corresponding author: Guanghui Hu

Received  September 2017 Revised  December 2017 Published  March 2018

Fund Project: The work of the first author is supported by the NSFC grant (No. 11671028), NSAF grant (No. U1530401) and the 1000-Talent Program of Young Scientists in China

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u = 0$ in $Q = (0,T)×Ω$ with $T>0$ and $Ω$ a $\mathcal C^2$ bounded domain of $\mathbb{R}^n$, $n≥2$. We start by considering the unique determination of some general singular time-dependent coefficients. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

Citation: Guanghui Hu, Yavar Kian. Determination of singular time-dependent coefficients for wave equations from full and partial data. Inverse Problems & Imaging, 2018, 12 (3) : 745-772. doi: 10.3934/ipi.2018032
##### References:
 [1] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar [2] M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. Google Scholar [3] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. Google Scholar [4] M. Bellassoued and I. Ben Aicha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Jour. Math. Anal. Appl., 449 (2017), 46-76. doi: 10.1016/j.jmaa.2016.11.082. Google Scholar [5] M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024. Google Scholar [6] M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745. Google Scholar [7] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873. Google Scholar [8] I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. Google Scholar [9] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247. Google Scholar [10] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Commun. Partial Diff. Eqns., 27 (2002), 653-668. doi: 10.1081/PDE-120002868. Google Scholar [11] P. Caro and K. M. Rogers, Global Uniqueness for The Calderón Problem with Lipschitz Conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28 pp. Google Scholar [12] M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, MCRF, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143. Google Scholar [13] M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (2017). doi: 10.1016/j.matpur.2017.12.003. Google Scholar [14] M. Choulli, Y. Kian and E. Soccorsi, Determining the time dependent external potential from the DN map in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558. doi: 10.1137/140986268. Google Scholar [15] M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, to appear Journal of Spectral Theory, arXiv: 1601.05355.Google Scholar [16] M. Choulli, Y. Kian and E. Soccorsi, On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974. doi: 10.1002/mma.4446. Google Scholar [17] F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715.Google Scholar [18] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005. Google Scholar [19] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758. doi: 10.1080/03605300701382340. Google Scholar [20] G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307. doi: 10.1007/s13373-017-0100-2. Google Scholar [21] D. D. Feirrera, C. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68. doi: 10.1080/03605302.2012.736911. Google Scholar [22] K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, MCRF, 6 (2016), 251-269. doi: 10.3934/mcrf.2016003. Google Scholar [23] P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. Google Scholar [24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. Google Scholar [25] B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3. Google Scholar [26] B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516. Google Scholar [27] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983. Google Scholar [28] V. Isakov, Completness of products of solutions and some inverse problems for PDE, J. Diff. Equat., 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A. Google Scholar [29] V. Isakov, An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195. doi: 10.1080/03605309108820794. Google Scholar [30] V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201. Google Scholar [31] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123 Chapman & Hall/CRC, Boca Raton, FL, 2001. Google Scholar [32] O. Kavian, Four lectures on parameter identification, three courses on partial differential equations, IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 4 (2003), 125-162. Google Scholar [33] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderon problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567. Google Scholar [34] Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732. doi: 10.3934/ipi.2014.8.713. Google Scholar [35] Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, Jour. Math. Anal. Appl., 436 (2016), 408-428. doi: 10.1016/j.jmaa.2015.12.018. Google Scholar [36] Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Annales de l'IHP (C) Nonlinear Analysis, 34 (2017), 973-990. doi: 10.1016/j.anihpc.2016.07.003. Google Scholar [37] Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046. doi: 10.1137/16M1076708. Google Scholar [38] Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, to appear in International Math Research Notices, available at https://doi.org/10.1093/imrn/rnx263.Google Scholar [39] Y. Kian, L. Oksanen and M. Morancey, Application of the boundary control method to partial data Borg-Levinson inverse spectral problem, preprint, arXiv: 1703.08832.Google Scholar [40] Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322.Google Scholar [41] I. Lasiecka, J.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. Google Scholar [42] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp. Google Scholar [43] J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. Google Scholar [44] Rakesh and A. G. Ramm, Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219. doi: 10.1016/0022-247X(91)90391-C. Google Scholar [45] Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Commun. Partial Diff. Eqns., 13 (1988), 87-96. doi: 10.1080/03605308808820539. Google Scholar [46] A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330. Google Scholar [47] R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. Google Scholar [48] P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158. Google Scholar [49] P. Stefanov and G. Uhlmann, Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, International Math Research Notices, 17 (2005), 1047-1061. Google Scholar [50] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690.Google Scholar [51] A. Waters, Stable determination of X-ray transforms of time dependent potentials from partial boundary data, Commun. Partial Diff. Eqns., 39 (2014), 2169-2197. doi: 10.1080/03605302.2014.930486. Google Scholar

show all references

##### References:
 [1] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar [2] M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. Google Scholar [3] M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. Google Scholar [4] M. Bellassoued and I. Ben Aicha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Jour. Math. Anal. Appl., 449 (2017), 46-76. doi: 10.1016/j.jmaa.2016.11.082. Google Scholar [5] M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024. Google Scholar [6] M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745. Google Scholar [7] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873. Google Scholar [8] I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. Google Scholar [9] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247. Google Scholar [10] A. L. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Commun. Partial Diff. Eqns., 27 (2002), 653-668. doi: 10.1081/PDE-120002868. Google Scholar [11] P. Caro and K. M. Rogers, Global Uniqueness for The Calderón Problem with Lipschitz Conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28 pp. Google Scholar [12] M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, MCRF, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143. Google Scholar [13] M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl. (2017). doi: 10.1016/j.matpur.2017.12.003. Google Scholar [14] M. Choulli, Y. Kian and E. Soccorsi, Determining the time dependent external potential from the DN map in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558. doi: 10.1137/140986268. Google Scholar [15] M. Choulli, Y. Kian and E. Soccorsi, Stability result for elliptic inverse periodic coefficient problem by partial Dirichlet-to-Neumann map, to appear Journal of Spectral Theory, arXiv: 1601.05355.Google Scholar [16] M. Choulli, Y. Kian and E. Soccorsi, On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data, Mathematical Methods in the Applied Sciences, 40 (2017), 5959-5974. doi: 10.1002/mma.4446. Google Scholar [17] F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715.Google Scholar [18] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005. Google Scholar [19] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758. doi: 10.1080/03605300701382340. Google Scholar [20] G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307. doi: 10.1007/s13373-017-0100-2. Google Scholar [21] D. D. Feirrera, C. Kenig and M. Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Comm. Partial Differential Equations, 38 (2013), 50-68. doi: 10.1080/03605302.2012.736911. Google Scholar [22] K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, MCRF, 6 (2016), 251-269. doi: 10.3934/mcrf.2016003. Google Scholar [23] P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. Google Scholar [24] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. Google Scholar [25] B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3. Google Scholar [26] B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516. Google Scholar [27] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983. Google Scholar [28] V. Isakov, Completness of products of solutions and some inverse problems for PDE, J. Diff. Equat., 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A. Google Scholar [29] V. Isakov, An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195. doi: 10.1080/03605309108820794. Google Scholar [30] V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201. Google Scholar [31] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123 Chapman & Hall/CRC, Boca Raton, FL, 2001. Google Scholar [32] O. Kavian, Four lectures on parameter identification, three courses on partial differential equations, IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 4 (2003), 125-162. Google Scholar [33] C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderon problem with partial data, Ann. of Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567. Google Scholar [34] Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732. doi: 10.3934/ipi.2014.8.713. Google Scholar [35] Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, Jour. Math. Anal. Appl., 436 (2016), 408-428. doi: 10.1016/j.jmaa.2015.12.018. Google Scholar [36] Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Annales de l'IHP (C) Nonlinear Analysis, 34 (2017), 973-990. doi: 10.1016/j.anihpc.2016.07.003. Google Scholar [37] Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046. doi: 10.1137/16M1076708. Google Scholar [38] Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, to appear in International Math Research Notices, available at https://doi.org/10.1093/imrn/rnx263.Google Scholar [39] Y. Kian, L. Oksanen and M. Morancey, Application of the boundary control method to partial data Borg-Levinson inverse spectral problem, preprint, arXiv: 1703.08832.Google Scholar [40] Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322.Google Scholar [41] I. Lasiecka, J.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. Google Scholar [42] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp. Google Scholar [43] J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968. Google Scholar [44] Rakesh and A. G. Ramm, Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219. doi: 10.1016/0022-247X(91)90391-C. Google Scholar [45] Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Commun. Partial Diff. Eqns., 13 (1988), 87-96. doi: 10.1080/03605308808820539. Google Scholar [46] A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330. Google Scholar [47] R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. Google Scholar [48] P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158. Google Scholar [49] P. Stefanov and G. Uhlmann, Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics, International Math Research Notices, 17 (2005), 1047-1061. Google Scholar [50] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690.Google Scholar [51] A. Waters, Stable determination of X-ray transforms of time dependent potentials from partial boundary data, Commun. Partial Diff. Eqns., 39 (2014), 2169-2197. doi: 10.1080/03605302.2014.930486. Google Scholar
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