On Some Mathematical Models of Filtration Theory

The article is devoted to the study of some mathematical models arising in filtration theory. We examine an inverse problem of determining an unknown right-hand side and coefficients in a pseudoparabolic equation of the third order. Equations of this type and more general Sobolev-type equations arise in filtration theory, heat and mass transfer, plasma physics, and in many other fields. We reduce the problem to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Together with the natural smoothness conditions for the data, we require also some well-posedness condition to be fulfilled which is actually reduced to the condition of nondegeneracy of some matrix constructed with the use of the data of the problem. Theorems on existence and uniqueness of solutions to this problem are stated and proven. Stability estimates are exposed. In the linear case the result is global in time, while in the nonlinear case it is local. The main function spaces used are the Sobolev spaces.

Keywords

pseudoparabolic equation; existence and uniqueness theorem; inverse problem; boundary value problem.

References

1. Lyubanova A.Sh., Tani A. On Inverse Problems for Pseudoparabolic and Parabolic Equations of Filtration. Inverse Problems in Science and Engineering, 2011, vol. 19, no. 7, pp. 1023-1042. DOI: 10.1080/17415977.2011.569712
2. Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner U.D. Linear and Non-Linear Sobolev-Type Equations. FML, 2007.
3. Bebernes J., Lacey A.A. Global Existence and Finite Time Blow-Up for a Class of Nonlocal Parabolic Problems. Adv. Differ. Equations, 1997, vol. 2, pp. 927-954.
4. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operator. Utrecht: VSP, 2003. 228 p. DOI: 10.1515/9783110915501
5. Chung S.K., Pani A.K. Numerical Methods for the Rosenau Equation. L. Appl. Anal., 2001, vol. 77, no. 3-4, pp. 100--116. DOI: 10.1080/00036810108840914
6. Chen Yu. Remark on the Global Existence for the Generalized Benjamin-Bona-Mahony Equations in Arbitrary Dimension. Appl. Anal., 1988, vol. 30, no. 1-3, pp. 1--15. DOI: 10.1080/00036818808839789
7. Egorov I.E., Pyatkov S.G., Popov S.V. Nonclassical Operator-Differential Equations. Novosibirsk, Nauka, 2000. 336 p.
8. Gajewski H. Groger K. Zacharias K. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Berlin, Akademie-Verlag, 1974.
9. Showalter R.E. Monotone Operators in Banach Space and Nonlinear Partial Differentail Equations. Providence, AMS, 1997. 278 p.
10. Kozhanov A.I. An Initial-boundary Value Problem for Equations of the Generalized Boussinesq Equation Type with a Nonlinear Source. Math. Notes, 1999, vol. 65, no. 1, pp. 59--63. DOI: 10.1007/BF02675010
11. Gladkov A.L. Unique Solvability of the Cauchy Problem for Certain Quasilinear Pseudoparabolic Equations. Math. Notes, 1996, vol. 60, no. 3, pp. 264-268. DOI: 10.1007/BF02320362
12. Di Benedetto E., Pierre M. On the Maximum Principle for Pseudoparabolic Equations. Indiana Univ. Math. J., 1981, vol. 30, no. 6, pp. 821-854. DOI: 10.1512/iumj.1981.30.30062
13. Begehr H., Dai D. Q. Initial Boundary Value Problem for Nonlinear Pseudoparabolic Equations. Complex Variables, Theory Appl., 1992, vol. 18, no. 1-2, pp. 33-47.
14. Mitidieri E., Pohozhaev S.I. A Priori Estimates and the Absence of Solutions of Nonlinear Equations and Inequalities of Partial. Proceedings of the Steklov Institute of Mathematics, 2001, vol. 234, no. 3, pp. 1-362.
15. Laptev G.G. About the Absence of Solutions for One Class of Singular Semilinear Differential Inequalities. Proceedings of the Steklov Institute of Mathematics, 2001, vol. 232, pp. 216-228.
16. Lyubanova A.Sh. Identification of a Coefficient in the Leading Term of a Pseudoparabolic Equation of Filtration. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 1048-1058. DOI: 10.1134/S0037446613060116
17. 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
18. Mamayusupov M.Sh. The Problem of Determining Coefficients of a Pseudoparabolic Equation. Studies in Integro-differential Equations, Ilim, Frunze, 1983, no. 16, pp. 290-297.
19. Favini A., Lorenzi A. Differential Equations. Inverse an Direct Problems. Tylor & Francis Group, LLC. 2006. DOI: 10.1201/9781420011135
20. Urazaeva A.V., Fedorov V.E. On the Well-Posedness of the Prediction-Control Problem for Certain Systems of Equations. Math. Notes, 2009, vol. 85, no. 3, pp. 426-436. DOI: 10.1134/S0001434609030134
21. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943276
22. Kabanikhin S.I. Inverse and Incorrect Problem. Novosibirsk: Siberian science publishing, 2009. 457 p. (in Russian)
23. Belov Ya.Ya. Inverse Problems for Parabolic Equations. Utrecht, VSP, 2002.
24. Ivanchov M. Inverse Problems for Equations of Parabolic Type. Math. Studies. Monograph Series. V. 10. Lviv, WNTL Publishers, 2003.
25. Isakov V. Inverse Problems for Partial Differential Equations. Berlin, Springer, 2006.
26. Prilepko A.I., Orlovsky D.G., and Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. N.Y., Marcel Dekker, Inc. 1999.
27. Pyatkov S.G. On Some Classes of Inverse Problems for Parabolic Equations. J. Inv. Ill-Posed Problems, 2011, vol. 18, no. 8, pp. 917-934.
28. Pyatkov S.G., Samkov M.L. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations. Sib. Adv. in Math., 2012, vol. 22, no. 4, pp. 287-302. DOI: 10.3103/S1055134412040050
29. Pyatkov S.G., Tsybikov B.N. On Some Classes of Inverse Problems for Parabolic and Elliptic Equations. J. Evol. Equat., 2011, vol. 11, no. 1, pp. 155-186. DOI: 10.1007/s00028-010-0087-6
30. Shergin S.N., Pyatkov S.G. On Some Classes of Inverse Problems for Pseudoparabolic Equations. Matem. Zam. SVFU., 2014. vol. 21, no. 2, pp. 106-116. (in Russian)
31. Gilbarg D., Trudinger N. Elliptic Differential Equation with Partial Derivative of the Second Order. Berlin, Heidelberg, Springer-Verlag, 2001.
32. Triebel H. Interpolation Theory. Function Space. Differential Operator. Berlin, VEB Deutscher Verlag der Wissenschaften, 1978. 528 p.
33. Krylov N.V. Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. AMS, 2008. DOI: 10.1090/gsm/096
34. Ladyzhenskaya O.A, Ural'tseva N.N. Linear and Quasilinear Elliptic Equations. Moscow: Nauka, 1964.
35. Amann H. Operator-Valued Foutier Multipliers, Vector-valued Besov Spaces and Applications. Mathem. Nachr., 1997, vol. 186, no. 1, pp. 5-56. DOI: 10.1002/mana.3211860102