# American Institute of Mathematical Sciences

June  2022, 15(6): 1439-1454. doi: 10.3934/dcdss.2022017

## Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements

 1 Dipartimento di Matematica, Universitá di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna BO, Italy 2 Department of Science and Engeneering, Sorbonne Unversity Abu Dhabi, Al Reem Island, 51133 Abu Dhabi, United Arab Emirates 3 Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via Edoardo Orabona 4, 70125 Bari BA, Italy

Received  September 2021 Revised  December 2021 Published  June 2022 Early access  February 2022

Let
 $\left(H, \langle \cdot, \cdot \rangle \right)$
be a separable Hilbert space and
 $A_{i}:D(A_i) \to H$
(
 $i = 1, \cdots, n$
) be a family of nonnegative selfadjoint operators mutually commuting. We study the inverse problem consisting in the identification of the function
 $u:[0, T] \to H$
and
 $n$
time-dependent diffusion coefficients
 $\alpha_{1}, \cdots, \alpha_{n}:[s, T] \to {\mathbb{R}}_+$
that fulfill the initial-value problem
 $u'(t) + \alpha_{1}(t) A_{1}u(t) + \cdots + \alpha_{n}(t) A_{n}u(t) = 0, \quad s \leq t \leq T, \quad u(s) = x,$
 $\langle A_{1} u(t), u(t)\rangle = \varphi_{1}(t), \quad \cdots \quad, \langle A_{n} u(t), u(t)\rangle = \varphi_{n}(t), \quad s \leq t \leq T.$
Under suitable assumptions on the operators
 $A_i$
,
 $i = 1, \cdots, n$
, on the initial data
 $x\in H$
and on the given functions
 $\varphi_i$
,
 $i = 1, \cdots, n$
, we shall prove that the solution of such a problem exists, is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion coefficients in a heat equation and of the Lamé parameters in an elasticity problem on a plate.
Citation: Angelo Favini, Gianluca Mola, Silvia Romanelli. Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1439-1454. doi: 10.3934/dcdss.2022017
##### References:
 [1] M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354. [2] K.-C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin and New York, 2005. [3] D. Huang, Y. Li and D. Pei, Identification of a time-dependent coefficient in heat conduction problem by new iteration method, Adv. Math. Phys., 2018 (2018), Art. ID 4918256, 7 pp. doi: 10.1155/2018/4918256. [4] M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publications, L'viv, Ukraine, 2003. [5] N. I. Ivanchov, On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity, Sib. Math. J., 35 (1994), 547-555.  doi: 10.1007/BF02104818. [6] N. I. Ivanchov and N. V. Pabyrivska, On determination of two time-dependent coefficients in a parabolic equation, Sib. Math. J., 43 (2002), 323-329.  doi: 10.1023/A:1014749222472. [7] T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234.  doi: 10.2969/jmsj/00520208. [8] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. [9] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in nonlinear differential equations and their application Vol. 16, Birkhauser, Basel-Boston-Berlin, 1995. doi: 10.1007/978-3-0348-9234-6. [10] G. Mola, Recovering a large number of diffusion constants in a parabolic equations from energy measurements, Inverse Problems and Imaging, 12 (2018), 527-543.  doi: 10.3934/ipi.2018023. [11] G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. doi: 10.1088/0266-5611/32/11/115016. [12] G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, Amer. J. Math., 115 (1993), 1161-1187.  doi: 10.2307/2375069. [13] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, M. Dekker, New York, 2000. [14] D. Trucu, D. B. Ingham and D. Lesnic, Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser., 124 (2008), 012050. [15] S. J. L. van Eijndhoven and J. de Graaf, A fundamental approach to the generalized eigenvalue problem for self-adjoint operators, J. Functional Analysis, 63 (1985), 74-85.  doi: 10.1016/0022-1236(85)90098-9.

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##### References:
 [1] M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354. [2] K.-C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin and New York, 2005. [3] D. Huang, Y. Li and D. Pei, Identification of a time-dependent coefficient in heat conduction problem by new iteration method, Adv. Math. Phys., 2018 (2018), Art. ID 4918256, 7 pp. doi: 10.1155/2018/4918256. [4] M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL Publications, L'viv, Ukraine, 2003. [5] N. I. Ivanchov, On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity, Sib. Math. J., 35 (1994), 547-555.  doi: 10.1007/BF02104818. [6] N. I. Ivanchov and N. V. Pabyrivska, On determination of two time-dependent coefficients in a parabolic equation, Sib. Math. J., 43 (2002), 323-329.  doi: 10.1023/A:1014749222472. [7] T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan, 5 (1953), 208-234.  doi: 10.2969/jmsj/00520208. [8] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. [9] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in nonlinear differential equations and their application Vol. 16, Birkhauser, Basel-Boston-Berlin, 1995. doi: 10.1007/978-3-0348-9234-6. [10] G. Mola, Recovering a large number of diffusion constants in a parabolic equations from energy measurements, Inverse Problems and Imaging, 12 (2018), 527-543.  doi: 10.3934/ipi.2018023. [11] G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. doi: 10.1088/0266-5611/32/11/115016. [12] G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, Amer. J. Math., 115 (1993), 1161-1187.  doi: 10.2307/2375069. [13] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, M. Dekker, New York, 2000. [14] D. Trucu, D. B. Ingham and D. Lesnic, Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser., 124 (2008), 012050. [15] S. J. L. van Eijndhoven and J. de Graaf, A fundamental approach to the generalized eigenvalue problem for self-adjoint operators, J. Functional Analysis, 63 (1985), 74-85.  doi: 10.1016/0022-1236(85)90098-9.
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