# Invariant Manifolds of Semilinear Sobolev Type Equations

O.G. KitaevaThe article is devoted to a review of the author's results in studying the stability of semilinear Sobolev type equations with a relatively bounded operator. We consider the initial-boundary value problems for the Hoff equation, for the Oskolkov equation of nonlinear fluid filtration, for the Oskolkov equation of plane-parallel fluid flow, for the Benjamin-Bon-Mahoney equation. Under an appropriate choice of function spaces, these problems can be considered as special cases of the Cauchy problem for a semilinear Sobolev type equation. When studying stability, we use phase space methods based on the theory of degenerate (semi)groups of operators and apply a generalization of the classical Hadamard-Perron theorem. We show the existence of stable and unstable invariant manifolds modeled by stable and unstable invariant spaces of the linear part of the Sobolev type equations in the case when the phase space is simple and the relative spectrum and the imaginary axis do not have common points.Full text

- Keywords
- Keywords: Sobolev type equations; invariant manifolds; Oskolkov equations; Hoff equation; Benjamin--Bon--Mahoney equation.
- References
- 1. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of ''Noises''. Abstract and Applied Analysis, 2015, article ID: 69741, 8 p. DOI: 10.1155/2015/697410

2. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of ''Noises''. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x

3. Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of ''Noises''. Electronic Journal of Differential Equations, 2018, vol. 2018, no. 128, pp. 1-10.

4. Hoff N.J. Creep Buckling. The Aeronautical Quarterly, 1956, vol. 7, no. 1, pp. 1-20.

5. Kitaeva O.G., Sviridyuk G.A. Stable and Unstable Invariant Manifolds of the Oskolkov Equation). International Seminar on Nonclassical Equations of Mathematical Physics Dedicated to the 60th Birth Anniversary of Professor Vladimir N. Vragov Novosibirsk, Russia, October 3-5, 2005, pp. 160-166. (in Russian)

6. Kitaeva O.G. Exponential Dichotomies of a Non-Classical Equations of Differential Forms on a Two-Dimensional Torus with ''Noises''. Journal of Computational and Engineering Mathematics, 2019, vol. 6, no. 3, pp. 26-38. DOI: 10.14529/jcem190303

7. Kitaeva O.G. Stable and Unstable Invariant Spaces of One Stochastic Non-Classical Equation with a Relatively Radial Operator on a 3-Torus. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 2, pp. 40-49. DOI: 10.14529/jcem200204

8. Kitaeva O.G., Shafranov D.E., Sviridyuk G.A. Degenerate Holomorphic Semigroups of Operators in Spaces of K- ''Noises'' on Riemannian Manifolds. Semigroups of Operators - Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics and Statistics. Springer, Cham, 2020, vol. 325, pp. 279-292. DOI: 10.1007978-3-030-46079-2_16

9. Kitaeva O.G. Exponential Dichotomies of a Stochastic Non-Classical Equation on a Two-Dimensional Sphere. Journal of Computational and Engineering Mathematics, 2021. vol. 8, no. 1, pp. 60-67. DOI: 10.14529/jcem210105

10. Kitaeva O.G. Invariant Spaces of Oskolkov Stochastic Linear Equations on the Manifold. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2021, vol. 13, no. 2, pp. 5-10. DOI: 10.14529/mmph210201

11. Kitaeva O.G. Invariant Manifolds of the Hoff Model in ''Noise''. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 4, pp. 24-35. DOI: 10.14529/mmp210402

12. Manakova N.A. An Optimal Control to Solutions of the Showalter-Sidorov Problem for the Hoff Model on the Geometrical Graph. Journal of Computational and Engineering Mathematics, 2014, vol. 1, no. 1, pp. 26-33.

13. Oskolkov A.P., Akhmatov M.M., Shchadiev R.D. Nonlocal Problems for Filtration Equations for Non-Newtonian Fluids in a Porous Medium. Journal of Soviet Mathematics, 1992, vol. 62, no. 5, pp. 2992-3004. DOI: 10.1007/BF01097498

14. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Journal of Mathematical Sciences, 1993, vol. 64, no. 1, pp. 724-736. DOI: 10.1007/BF02988478

15. Oskolkov A.P. On Stability Theory for Solutions of Semilinear Dissipative Equations of the Sobolev Type. Journal of Mathematical Sciences, 1995, vol. 77, no. 3, pp. 3225-3231. DOI: 10.1007/BF02364715

16. Sagadeeva M.A., Zagrebina S.A., Manakova N.A. Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation. Evolution Equations and Control Theory, 2019, vol. 8, no. 3, pp. 473-488. DOI: 10.3934/eect.2019023

17. Shafranov D.E. Stability of Solutions for the Linear Oskolkov System in the k-Form Spaces Defined on the Riemannian Manifold. Bulletin of the Samara State University: Natural Science Series, 2007, vol. 56, no. 6, pp. 155-161. (in Russian)

18. Shafranov D.E., Shvedchikova A.I. The Hoff Equation as a Model of Elastic Shell. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 18 (277), issue 12, pp. 77-81. (in Russian)

19. Shafranov D.E. Numeral Solution of the Barenblatt-Zheltov-Kochina Equation with Additive ''White Noise'' in Spaces of Differential Forms on a Torus. Journal of Computational and Engineering Mathematics, 2019, vol. 6, no. 4, pp. 31-43. DOI: 10.14529/jcem190403

20. Shafranov D.E. Numeral Solution of the Dzektser Equation with ''White Noise'' in Space of Smooth Differential Forms Defined on a Torus. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 2, pp. 58-65. DOI: 10.14529/jcem200206

21. Shafranov D.E. Numerical Solution of the Hoff Equation with Additive ''White Noise'' in Spaces of Differential Forms on a Torus. Journal of Computational and Engineering Mathematics, 2021, vol. 8, no. 2, pp. 46-55. DOI: 10.14529/jcem210204

22. Shafranov D.E., Kitaeva O.G., Sviridyuk G.A. Stochastic Equations of Sobolev Type with Relatively p-Radial Operators in Spaces of Differential Forms. Differential Equations, 2021, vol. 57, no. 4, pp. 507-516. DOI: 10.1134/S0012266121040078

23. Shestakov A.L., Sviridyuk G.A., Hudyakov Yu.V. Dinamic Measurement in Spaces of ''Noises''. Bulletin of the South Ural State University. Series: Computer Technology, Management, Radio Electronics, 2013, vol. 13, no. 2, pp. 4-11. (in Russian)

24. Sviridyuk G.A. [The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation]. Dokldy Akademii Nauk SSSR, 1986, vol. 289, no. 6, pp. 1315-1318. (in Russian)

25. Sviridyuk G.A. On the Variety of Solutions of a Certain Problem of an Incompressible Viscoelastic Fluid. Differential Equations, 1988, vol. 24, no. 10, pp. 1846-1848.

26. Sviridyuk G.A. A Problem for the Generalized Boussinesq Filtration Equation. Soviet Mathematics (Izvestiya VUZ. Matematika), 1989, vol. 33, no. 2, pp. 62-73.

27. Sviridyuk G.A. Solvability of a Problem of the Thermoconvection of a Viscoelastic Incompressible Fluid. Soviet Mathematics (Izvestiya VUZ. Matematika), 1990, vol. 34, no. 12, pp. 80-86.

28. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russian Mathematical Surveys, 1994, vol. 49, no. 4, pp. 45-74. DOI: 10.1070/RM1994v049n04ABEH002390

29. Sviridyuk G.A., Yakupov M.M. The Phase Space of the Initial-Boundary Value Problem for the Oskolkov System. Differential Equations, 1996, vol. 32, no. 11, pp. 1535-1540.

30. Sviridyuk G.A., Keller A.V. Invariant Spaces and Dichotomies of Solutions of a Class of Linear Equations of Sobolev Type. Russian Mathematics, 1997, vol. 41, no. 5, pp. 57-65.

31. Sviridyuk G.A., Efremov A.A. Optimal Control of a Class of Linear Degenerate Equations. Doklady Mathematics, 1999, vol. 59, no. 1, pp. 157-159.

32. Sviridyuk G.A., Manakova N.A. The Phase Space of the Cauchy-Dirichlet Problem for the Oskolkov Equation of Nonlinear Filtration. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, no. 9, pp. 33-38.

33. Sviridyuk G.A., Ankudinov A.V. The Phase Space of the Cauchy-Dirichlet Problem for a Nonclassical Equation. Differential Equations, 2003, vol. 39, no. 11, pp. 1639-1644. DOI: 10.1023/B:DIEQ.0000019357.68736.15

34. Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 2, pp. 262-266. DOI: 10.4213/mzm347

35. Sviridyuk G.A., Kitaeva O.G. Invariant Manifolds of the Hoff Equation. Mathematical Notes, 2006, vol. 79, no. 3, pp. 408-412. DOI: 10.4213/mzm2713

36. Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145. DOI: 10.1134/S0012266106010125

37. Sviridyuk G.A., Manakova N.A. An Optimal Control Problem for the Hoff Equation. Journal of Applied and Industrial Mathematics, 2007, vol. 1, no. 2, pp. 247-253. DOI: 10.1134/S1990478907020147

38. Sviridyuk G.A., Manakova N.A. The Barenblatt-Zheltov-Kochina Model with Additive White Noise in Quasi-Sobolev Spaces. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 1, pp. 61-67. DOI: 10.14529/jcem16010

39. Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier-Stokes Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132-136. DOI: 10.14529/mmp150111

40. Zamyshlyaeva A.A., Bychkov E.V., Tsyplenkova O.N. Mathematical Models Based on Boussinesq-Love Equation. Applied Mathematical Sciences, 2014, vol. 8, pp. 5477-5483. DOI: 10.12988/ams.2014.47546

41. Zamyshlyaeva A.A., Al-Isawi J.K.T. On Some Properties of Solutions to One Class of Evolution Sobolev Type Mathematical Models in Quasi-Sobolev Spaces. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 4, pp. 113-119. DOI: 10.14529/mmp150410

42. Zamyshlyaeva A.A., Lut A.V. Numerical Investigation of the Boussinesq-Love Mathematical Models on Geometrical Graphs. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 2, pp. 137-143. DOI: 10.14529/mmp170211

43. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Sobolev Type Equation of Higher Order. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 1, pp. 5-14. DOI: 10.14529/mmp180101