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 and Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
References:
[1]

S. A. AvdoninM. 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.

[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.

[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.

[4]

C. Bennewitz, A proof of the local Borg-Marchenko theorem, Communications of Mathematical Physics, 218 (2001), 131-132.  doi: 10.1007/s002200100384.

[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). 

[6]

A. L. BukhgeimN. 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.

[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.

[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.

[9]

S. Ivanov, Singularity Propagation for the Gurtin-Pipkin Equation, arXiv: 1312.1580

[10]

S. Ivanov, Regularity of the Gurtin-Pipkin equation, http://arXiv.org/abs/1205.0616

[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.

[12]

D. D. JosephA. Narain and O. Riccius, Shear-wave speeds and elastic moduli for different liquids. Part 1. Theory, J Fluid Mech., 171 (1976), 1289-1308. 

[13]

B. M. Levitan, Inverse Sturm-Liouville Problems, Utrecht: The Netherlands, 1987.

[14]

V. A. Marchenko, Certain problems in the theory of second-order differential operators, Doklady Akad. Nauk SSSR, 72 (1950), 457-460 (Russian). 

[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.

[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.

[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.

[18]

B. Simon, A new aproach to inverse spectral theory, I. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.  doi: 10.2307/121061.

[19]

V. V. VlasovN. 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.

show all references

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

References:
[1]

S. A. AvdoninM. 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.

[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.

[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.

[4]

C. Bennewitz, A proof of the local Borg-Marchenko theorem, Communications of Mathematical Physics, 218 (2001), 131-132.  doi: 10.1007/s002200100384.

[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). 

[6]

A. L. BukhgeimN. 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.

[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.

[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.

[9]

S. Ivanov, Singularity Propagation for the Gurtin-Pipkin Equation, arXiv: 1312.1580

[10]

S. Ivanov, Regularity of the Gurtin-Pipkin equation, http://arXiv.org/abs/1205.0616

[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.

[12]

D. D. JosephA. Narain and O. Riccius, Shear-wave speeds and elastic moduli for different liquids. Part 1. Theory, J Fluid Mech., 171 (1976), 1289-1308. 

[13]

B. M. Levitan, Inverse Sturm-Liouville Problems, Utrecht: The Netherlands, 1987.

[14]

V. A. Marchenko, Certain problems in the theory of second-order differential operators, Doklady Akad. Nauk SSSR, 72 (1950), 457-460 (Russian). 

[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.

[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.

[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.

[18]

B. Simon, A new aproach to inverse spectral theory, I. Fundamental formalism, Ann. of Math., 150 (1999), 1029-1057.  doi: 10.2307/121061.

[19]

V. V. VlasovN. 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.

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