# American Institute of Mathematical Sciences

February  2019, 13(1): 31-38. doi: 10.3934/ipi.2019002

## Inverse problems for the heat equation with memory

 1 University of Alaska, Fairbanks, AK 99775-6660, USA 2 Russian Academy of Sciences, St. Petersburg, Mendeleev line 1, Russia 3 Beijing Institute of Technology, Beijing 100081, China

The third author is supported by the National Natural Science Foundation of China, grant 61673061.

Received  August 2017 Revised  October 2018 Published  December 2018

Fund Project: The first author is supported by the NSF, grant DMS 1411564, and by the Ministry of Education and Science of Republic of Kazakhstan under the grant no. AP05136197

We study inverse boundary problems for one dimensional linear integro-differential equation of the Gurtin-Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis t>0. For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg-Marchenko theorem for the Schrödinger equation.

Citation: Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
##### References:
 [1] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Matrix inverse problem for the equation $u_{tt} - u_{xx} + Q(x)u = 0$, Math. USSR Sbornik, 72 (1992), 287-310. doi: 10.1070/SM1992v072n02ABEH002141. Google Scholar [2] S. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009. Google Scholar [3] S. Avdonin and L. Pandolfi, A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements, J. Inverse Ill-Posed Probl., 26 (2018), 299-310. doi: 10.1515/jiip-2016-0064. Google Scholar [4] C. Bennewitz, A proof of the local Borg-Marchenko theorem, Communications of Mathematical Physics, 218 (2001), 131-132. doi: 10.1007/s002200100384. Google Scholar [5] A. S. Blagoveschenskii, On a local approach to the solution of the dynamical inverse problem for an inhomogeneous string, Trudy MIAN, 115 (1971), 28-38 (in Russian). Google Scholar [6] A. L. Bukhgeim, N. I. Kalinina and V. B. Kardakov, Two methods in an inverse problem of memory reconstruction, Siberian Math. J., 41 (2000), 767-776. doi: 10.1007/BF02679689. Google Scholar [7] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Archive for Rational Mechanics and Analysis, 32 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar [8] S. A. Ivanov and A. Eremenko, Spectra of the Gurtin-Pipkin type equations, SIAM J. Math. Anal., 43 (2011), 2296-2306. doi: 10.1137/100811908. Google Scholar [9] S. Ivanov, Singularity Propagation for the Gurtin-Pipkin Equation, arXiv: 1312.1580Google Scholar [10] S. Ivanov, Regularity of the Gurtin-Pipkin equation, http://arXiv.org/abs/1205.0616Google Scholar [11] J. Janno and L. von Wolfersdorf, Inverse problems for memory kernels by Laplace transform methods, Zeitschrift fur Analysis und ihre Anwendungen, 19 (2000), 489-510. doi: 10.4171/ZAA/963. Google Scholar [12] D. D. Joseph, A. Narain and O. Riccius, Shear-wave speeds and elastic moduli for different liquids. Part 1. Theory, J Fluid Mech., 171 (1976), 1289-1308. Google Scholar [13] B. M. Levitan, Inverse Sturm-Liouville Problems, Utrecht: The Netherlands, 1987. Google Scholar [14] V. A. Marchenko, Certain problems in the theory of second-order differential operators, Doklady Akad. Nauk SSSR, 72 (1950), 457-460 (Russian). Google Scholar [15] L. Pandolfi, A linear algorithm for the identification of a relaxation kernel using two boundary measures, Inverse Problems, 31 (2015), 105003, 12 pp. doi: 10.1088/0266-5611/31/10/105003. Google Scholar [16] L. Pandolfi, Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution, Math. Methods Appl. Sci., 40 (2017), 2542-2549. doi: 10.1002/mma.4180. Google Scholar [17] L. Pandolfi, Distributed Systems with Persistent Memory Control and Moment Problems, Springer. Briefs in Electrical and Computer Engineering, 2014. doi: 10.1007/978-3-319-12247-2. Google Scholar [18] B. Simon, A new aproach to inverse spectral theory, I. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057. doi: 10.2307/121061. Google Scholar [19] V. V. Vlasov, N. A. Rautian and A. S. Shamaev, Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics, Journal of Mathematical Sciences, 190 (2013), 34-65. doi: 10.1007/s10958-013-1245-5. Google Scholar

show all references

##### References:
 [1] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Matrix inverse problem for the equation $u_{tt} - u_{xx} + Q(x)u = 0$, Math. USSR Sbornik, 72 (1992), 287-310. doi: 10.1070/SM1992v072n02ABEH002141. Google Scholar [2] S. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009. Google Scholar [3] S. Avdonin and L. Pandolfi, A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements, J. Inverse Ill-Posed Probl., 26 (2018), 299-310. doi: 10.1515/jiip-2016-0064. Google Scholar [4] C. Bennewitz, A proof of the local Borg-Marchenko theorem, Communications of Mathematical Physics, 218 (2001), 131-132. doi: 10.1007/s002200100384. Google Scholar [5] A. S. Blagoveschenskii, On a local approach to the solution of the dynamical inverse problem for an inhomogeneous string, Trudy MIAN, 115 (1971), 28-38 (in Russian). Google Scholar [6] A. L. Bukhgeim, N. I. Kalinina and V. B. Kardakov, Two methods in an inverse problem of memory reconstruction, Siberian Math. J., 41 (2000), 767-776. doi: 10.1007/BF02679689. Google Scholar [7] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Archive for Rational Mechanics and Analysis, 32 (1968), 113-126. doi: 10.1007/BF00281373. Google Scholar [8] S. A. Ivanov and A. Eremenko, Spectra of the Gurtin-Pipkin type equations, SIAM J. Math. Anal., 43 (2011), 2296-2306. doi: 10.1137/100811908. Google Scholar [9] S. Ivanov, Singularity Propagation for the Gurtin-Pipkin Equation, arXiv: 1312.1580Google Scholar [10] S. Ivanov, Regularity of the Gurtin-Pipkin equation, http://arXiv.org/abs/1205.0616Google Scholar [11] J. Janno and L. von Wolfersdorf, Inverse problems for memory kernels by Laplace transform methods, Zeitschrift fur Analysis und ihre Anwendungen, 19 (2000), 489-510. doi: 10.4171/ZAA/963. Google Scholar [12] D. D. Joseph, A. Narain and O. Riccius, Shear-wave speeds and elastic moduli for different liquids. Part 1. Theory, J Fluid Mech., 171 (1976), 1289-1308. Google Scholar [13] B. M. Levitan, Inverse Sturm-Liouville Problems, Utrecht: The Netherlands, 1987. Google Scholar [14] V. A. Marchenko, Certain problems in the theory of second-order differential operators, Doklady Akad. Nauk SSSR, 72 (1950), 457-460 (Russian). Google Scholar [15] L. Pandolfi, A linear algorithm for the identification of a relaxation kernel using two boundary measures, Inverse Problems, 31 (2015), 105003, 12 pp. doi: 10.1088/0266-5611/31/10/105003. Google Scholar [16] L. Pandolfi, Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution, Math. Methods Appl. Sci., 40 (2017), 2542-2549. doi: 10.1002/mma.4180. Google Scholar [17] L. Pandolfi, Distributed Systems with Persistent Memory Control and Moment Problems, Springer. Briefs in Electrical and Computer Engineering, 2014. doi: 10.1007/978-3-319-12247-2. Google Scholar [18] B. Simon, A new aproach to inverse spectral theory, I. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057. doi: 10.2307/121061. Google Scholar [19] V. V. Vlasov, N. A. Rautian and A. S. Shamaev, Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics, Journal of Mathematical Sciences, 190 (2013), 34-65. doi: 10.1007/s10958-013-1245-5. Google Scholar
 [1] Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015 [2] Luciano Pandolfi. Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-11. doi: 10.3934/dcdss.2020090 [3] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [4] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [5] Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165 [6] Michel Pierre, Morgan Pierre. Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5347-5377. doi: 10.3934/dcds.2013.33.5347 [7] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [8] Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509 [9] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [10] 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 [11] Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007 [12] Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 [13] 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 [14] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [15] Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029 [16] Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465 [17] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [18] Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems & Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731 [19] Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073 [20] Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

2018 Impact Factor: 1.469