Inverse Problems for Mathematical Models of Quasistationary Electromagnetic Waves in Anisotropic Nonmetallic Media with Dispersion

We consider inverse problems of evolution type for mathematical models of quasistationary electromagnetic waves. It is assumed in the model that the wave length is small as compared with space inhomogeneities. In this case the electric and magnetic potential satisfy elliptic equations of second order in the space variables comprising integral summands of convolution type in time. After differentiation with respect to time the equation is reduced to a composite type equation with an integral summand. The boundary conditions are supplemented with the overdetermination conditions which are a collection of functionals of a solution (integrals of a solution with weight, the values of a solution at separate points, etc.). The unknowns are a solution to the equation and unknown coefficients in the integral operator. Global (in time) existence and uniqueness theorems of this problem and stability estimates are established.

Keywords

Sobolev-type equation; equation with memory; elliptic equation; inverse problem; boundary value problem.

References

1. Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner U.D. Lineynye i nelineynye uravneniya sobolevskogo tipa [Linear and Non-Linear Sobolev Equations]. Moscow, Fizmatlit, 2007. (in Russian)
2. Gabov S.A., Sveshnikov A.G. Lineynye zadachi teorii nestatsionarnykh vnutrennikh voln [Linear Problems of the Theory of Nonstationary Interior Waves]. Moscow, Nauka, 1990. (in Russian)
3. Lorenzi A., Paparone I. Direct and Inverse Problems in the Theory of Materials with Memory. Rendiconti del Seminario Matematico della Universita di Padova, 1992, vol. 87, p. 105-138.
4. Janno J., Von Wolfersdorf L. Inverse Problems for Identification of Memory Kernels in Viscoelasticity. Mathematical Methods in the Applied Sciences, 1997, vol. 20, pp. 291-314. DOI: 10.1002/(SICI)1099-1476(19970310)20:4<291::AID-MMA860>3.0.CO;2-W
5. Durdiev D.K., Safarov Zh.Sh. Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain. Mathematical Notes, 2015, vol. 97, no. 6, pp. 867-877. DOI: 10.1134/S0001434615050223
6. Colombo F., Guidetti D. An Inverse Problem for a Phase-Field Model in Sobolev Spaces. Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, Basel, Birkh'auser Verlag, 2005, vol. 64, pp. 189-210.
7. Guidetti D., Lorenzi A. A Mixed Type Identification Problem Related to a Phase-Field Model with Memory. Osaka Journal of Mathematics, 2007, vol. 44, pp. 579-613.
8. Colombo F., Guidetti D. A Global in Time Existence and Uniqueness Result for a Semilinear Integrodifferential Parabolic Inverse Problem in Sobolev Spaces. Mathematical Models and Methods in Applied Sciences, 2007, vol. 17, no. 4, pp. 537-565. DOI: 10.1142/S0218202507002017
9. Colombo F. On Some Methods to Solve Integro-Differential Inverse Problems of Parabolic Type. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 95-115.
10. Favini A., Lorenzi A. Identication Problems for Singular Integro-Differential Equations of Parabolic Type. Nonlinear Analysis, 2004, vol. 56, no. 6, pp. 879-904. DOI: 10.1016/j.na.2003.10.018
11. Lorenzi A., Tanabe H. Inverse and Direct Problems for Nonautonomous Degenerate Integro-Differential Equations of Parabolic Type with Dirichlet Boundary Conditions. Differential Equations: Inverse and Direct Problems. Lecture Notes in Pure and Applied Mathematics, Boca Raton, London, N.Y., Chapman and Hall/CRC Taylor and Francis Group, 2006, vol. 251, pp. 197-244.
12. Abaseeva N., Lorenzi A. Identification Problems for Nonclassical Integro-Differential Parabolic Equations. Journal of Inverse and Ill-Posed Problems, 2005, vol. 13, no. 6, pp. 513-535. DOI: 10.1515/156939405775199523
13. Asanov A., Atamanov E.R. An Inverse Problem for a Pseudoparabolic Integro-Defferential Operator Equation. Siberian Mathematical Journal, 1995, vol. 38, no. 4, pp. 645-655. DOI: 10.1007/BF02107322
14. Avdonin S.A., Ivanov S.A., Wang J. Inverse Problems for the Heat Equation with Memory, 2017, 10 p. Available at: https://arxiv.org/abs/1612.02129 (accessed February 09, 2018).
15. Pandolfi L. Identification of the Relaxation Kernel in Diffusion Processes and Viscoelasticity with Memory via Deconvolution, 2016, 15 p. Available at: https://arxiv.org/abs/1603.04321 (accessed February 09, 2018).
16. Denisov A.M. An Inverse Problem for a Quasilinear Integro-Differential Equation. Differential Equations, 2001, vol. 37, no. 10, pp. 1420-1426. DOI: 10.1023/A:1013320315508
17. Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin: VEB Deutscher Verlag der Wissenschaften, 1978.
18. Ladyzhenskaya O.A, Ural'tseva N.N. Linear and Quasilinear Elliptic Equations. N.Y., Academic Press, 2016.
19. Gilbarg D., Trudinger N. Ellipticheskie differentsial'nye uravneniya s chastnymi proizvodnymi vtorogo poryadka [Elliptic Differential Equation with Partial Derivative of the Second Order]. Moscow, Nauka, 1989.
20. Maugeri A., Palagachev D.K., Softova L.G. Elliptic and Parabolic Equations with Discontinuous Coefficients. Berlin, Wiley-VCH Verlag, 2000. DOI: 10.1002/3527600868