On Some Methods to Solve Integrodifferential Inverse Problems of Parabolic Type

В статье приводится обзор некоторых методов решения класса интегро-дифференциальных обратных задач. А именно, представлены методы решения интегро-дифференциальных обратных задач параболического типа, основанные на теории аналитических полугрупп, результатах по оптимальной регулярности и теореме о неподвижной точке. Эти методы могут быть применены к большому классу физических моделей, например, модели фазового поля, моделей горения, модели сильно затухающих волн с памятью.

Ключевые слова

интегро-дифференциальные обратные задачи; аналитические полугруппы.

Литература

1. Colombo F., Lorenzi A. Identification of Time and Space Dependent Relaxation Kernels in the Theory of Materials with Memory I. Math. Anal. Appl., 1997, no. 213, pp. 32-62.
2. Colombo F., Lorenzi A. Identification of Time and Space Dependent Relaxation Kernels in the Theory of Materials with Memory II. J. Math. Anal. Appl., 1997, no. 213, pp. 63-90. DOI: 10.1006/jmaa.1997.5365
3. Colombo F., Lorenzi A. An Identification Problem Related to a Parabolic Integrodifferential Equation with Noncommuting Spatial Operators. J. Inverse Ill-Posed Probl., 2000, no. 8, pp. 505-540.
4. Colombo F., Guidetti D., Lorenzi A. Integrodifferential Identification Problems for the Heat Equation in Cylindrical Domains. Adv. Math. Sci. Appl., 2003, no. 13, pp. 639-662.
5. Colombo F., Guidetti D., Lorenzi A. Integrodifferential Identification Problems for Thermal Materials with Memory in Non-Smooth Plane Domains. Dyn. Syst. Appl., 2003, no. 12, pp. 533-560.
6. Lorenzi A. Direct and Inverse Integrodifferential Maxwell Problems for Dispersive Media Related to Cylindrical Domains. Philadelphia, SIAM, 1995.
7. Di Cristo M., Guidetti D., Lorenzi A. Abstract Parabolic Equations with Applications to Problems in Cylindrical Space Domains. Adv. Differential Equations, 2010, no. 15, pp. 1-42.
8. Colombo F. Direct and Inverse Problems for a Phase-Field Model with Memory. J. Math. Anal. Appl., 2001, no. 260, pp. 517-545. DOI: 10.1006/jmaa.2001.7475
9. Lorenzi A., Rocca E. Identification of Two Memory Kernels in a Fully Hyperbolic Phase-Field System. J. Inverse Ill-Posed Probl., 2008, no. 16, pp. 147-174. DOI: 10.1515/jiip.2008.010
10. Lorenzi A. An Identification Problem for a Conserved Phase-Field Model with Memory. Math. Methods Appl. Sci., 2005, no. 28, pp. 1315-1339. DOI: 10.1002/mma.614
11. Guidetti D., Lorenzi A. A Mixed Type Identification Problem Related to a Phase-field Model with Memory. Osaka J. Math., 2007, no. 44, pp. 579-613.
12. Colombo F., Guidetti D., Lorenzi A. Identification of Two Memory Kernels and the Time Dependence of the Heat Source for a Parabolic Conserved Phase-Field Model. Math. Meth. Appl. Sci., 2006, no. 28, pp. 2085-2115.
13. Colombo F., Lorenzi A. An Inverse Problem in the Theory of Combustion of Materials with Memory. Adv. Differential Equations, 1998, no. 3, pp. 133-154.
14. Colombo F. An Inverse Problem for a Parabolic Integrodifferential Model in the Theory of Combustion. Physica D, 2007, no. 236, pp. 81-89. DOI: 10.1016/j.physd.2007.07.012
15. Colombo F., Guidetti D. A Unified Approach to Nonlinear Integrodifferential Inverse Problems of Parabolic Type. Z. Anal. Anwendungen, 2002, no. 21, pp. 431-464.
16. Colombo F. An Inverse Problem for the Strongly Damped Wave Equation with Memory. Nonlinearity, 2007, no. 20, pp. 659-683. DOI: 10.1088/0951-7715/20/3/006
17. Colombo F., Guidetti D. Identification of the Memory Kernel in the Strongly Damped Wave Equation by a Flux Condition. Commun. Pure Appl. Anal., 2009, no. 8, pp. 601-620.
18. Bakushinsky A.B., Kokurin M.Yu. Iterative Methods for Approximate Solution of Inverse Problems. Dordrecht, Springer, 2004.
19. Bertero M., Boccacci P. Introduction to Inverse Problems in Imaging. Bristol, Institute of Physics Publishing, 1998. DOI: 10.1887/0750304359
20. Groetsch C.W. Inverse Problems. Activities for Undergraduates. Washington, Mathematical Association of America, 1999.
21. Gladwell G.M.L. Inverse Problems in Vibration. Dordrecht, Kluwer Academic Publishers, 2004.
22. Kabanikhin S.I., Lorenzi A. Identification Problems of Wave Phenomena, Theory and Numerics. Utrecht, VSP, 1999.
23. Isakov V. Inverse Problems for Partial Differential Equations. New York, Springer-Verlag, 1998. DOI: 10.1007/978-1-4899-0030-2
24. Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems. New York, Springer-Verlag, 1996. DOI: 10.1007/978-1-4612-5338-9
25. Kaipio J., Somersalo E. Statistical and Computational Inverse Problems. New York, Springer-Verlag, 2005.
26. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. New York, Marcel Dekker, Inc., 2000.
27. Romanov V.G. Investigation Methods for Inverse Problems. Utrecht, VSP, 2002. DOI: 10.1515/9783110943849
28. Colombo F., Guidetti D. An Inverse Problem for a Phase-Field Model in Sobolev Spaces. Progress in Nonlinear Differential Equations and Their Applications, 2005, no. 64, pp. 189-210.
29. Colombo F., Guidetti D. A Global in Time Existence and Uniqueness Result for a Semilinear Integrodifferential Parabolic Inverse Problem in Sobolev Spaces. Math. Models Methods Appl. Sci., 2007, no. 17, pp. 1-29. DOI: 10.1142/s0218202507002017
30. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Basel, Birkhauser Verlag, 1995.
31. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Berlin, Deutscher Verlag der Wissenschaften, 1978.
32. Triebel H. Theory of Functions Spaces. Basel, Birkhauser Verlag, 1983. DOI: 10.1007/978-3-0346-0416-1
33. Adams R. Sobolev Spaces. New York, Academic Press, 1975.
34. Dore G. Maximal Regularity in L^p Spaces for an Abstract Cauchy Problem. Adv. Diff. Eq., 2000, no. 5, pp. 293-322.
35. Hillen T. Qualitative Analysis of Semilinear Cattaneo Equations. Math. Models Methods Appl. Sci., 1998, no. 8, pp. 507-519. DOI: 10.1142/S0218202598000238