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. BellassouedM. 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. BellassouedD. 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. ChoulliY. 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. ChoulliY. 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. FeirreraC. 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. KenigJ. 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. LasieckaJ.-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. BellassouedM. 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. BellassouedD. 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. ChoulliY. 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. ChoulliY. 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. FeirreraC. 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. KenigJ. 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. LasieckaJ.-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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