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

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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

Guanghui Hu ^1,, and Yavar Kian ^2,

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.

Keywords: Inverse problems, wave equation, time dependent coefficient, singular coefficients, Carleman estimate.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35L05.

Citation: Guanghui Hu, Yavar Kian. Determination of singular time-dependent coefficients for wave equations from full and partial data. Inverse Problems and 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.
[2]	M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.
[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.
[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.
[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.
[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.
[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.
[8]	I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp.
[9]	A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715.
[18]	G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005.
[19]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.
[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.
[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.
[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.
[23]	P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp.
[24]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.
[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.
[26]	B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516.
[27]	L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983.
[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.
[29]	V. Isakov, An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195. doi: 10.1080/03605309108820794.
[30]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322.
[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.
[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.
[43]	J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968.
[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.
[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.
[46]	A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.
[47]	R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp.
[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.
[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.
[50]	P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690.
[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.

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.
[2]	M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.
[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.
[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.
[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.
[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.
[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.
[8]	I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp.
[9]	A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, Sov. Math.-Dokl., 24 (1981), 244-247.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	F. Chung and L. Tzou, The $L^p$ Carleman estimate and a partial data inverse problem, preprint, arXiv: 1610.01715.
[18]	G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005.
[19]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Commun. Partial Diff. Eqns., 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.
[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.
[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.
[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.
[23]	P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp.
[24]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.
[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.
[26]	B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. Journal, 162 (2013), 496-516.
[27]	L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol Ⅲ, Springer-Verlag, Berlin, Heidelberg, 1983.
[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.
[29]	V. Isakov, An inverse hyperbolic problem with many boundary measurements, Commun. Partial Diff. Eqns., 16 (1991), 1183-1195. doi: 10.1080/03605309108820794.
[30]	V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rat. Mech. Anal., 124 (1993), 1-12. doi: 10.1007/BF00392201.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, preprint, arXiv: 1705.01322.
[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.
[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.
[43]	J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris, 1968.
[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.
[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.
[46]	A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.
[47]	R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp.
[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.
[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.
[50]	P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, preprint, arXiv: 1607.08690.
[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.

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