Exponential Dichotomies in Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with 'Noise'

Исследована устойчивость решений в линейных стохастических моделях соболевского типа с относительно ограниченным оператором в пространствах гладких дифференциальных форм, определенных на гладких компактных ориентированных римановых многообразиях без края. Для этого в пространстве дифференциальных форм используем вместо обычного оператора Лапласа псевдодифференциальный оператор Лапласа - Бельтрами. В качестве начальных использованы условие Коши и условие Шоуолтера - Сидорова. В связи с недифферинцируемостью, в обычном понимании, имеющегося в модели 'белого шума' используем производную стохастического процесса в смысле Нельсона - Гликлиха. Для исследования устойчивости решений устанавливаем наличие экспоненциальных дихотомий разделяющих пространство решений на устойчивое и неустойчивое инвариантные подпространства. В качестве примера используется стохастический вариант уравнения Баренблатта - Желтова - Кочиной в пространстве дифференциальных форм, определенных на гладком компактном ориентированном римановом многообразии без края.

Ключевые слова

уравнения соболевского типа; дифференциальные формы; стохастические уравнения; производная Нельсона - Гликлиха.

Литература

1. 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
2. Shafranov D.E. The Splitting of the Domain of the Definition of the Elliptic Self-Adjoint Pseudodifferential Operator. Journal of Computation and Engineering Mathematics, 2015, vol. 2, no. 3, pp. 60-64. DOI: 10.14529/jcem150306
3. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolv Type with Showalter-Sidorov Condition and Additive 'Noise'. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. DOI: 10.14529/mmp140108 (in Russian)
4. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer, 2011. DOI: 10.1007/978-0-85729-163-9
5. 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
6. 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
7. 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.
8. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive 'White Noise'. Communications on Pure and Applied Analysis, 2016, vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
9. Shafranov D.E., Kitaeva O.G. The Barenblatt-Zheltov-Kochina Model with the Showalter-Sidorov Condition and Additive 'White Noise' in Spaces of Differential Forms on Riemannian Manifolds without Boundary. Global and Stochastic Analysis, 2018, vol. 5, no. 2, pp. 145-159.
10. 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.
11. Melnikova I.V., Alshanskiy M.A. Generalized Solutions of Abstract Stochastic Problems. Pseudo-Differential Operators, Generalized Functions and Asymptotics, 2013, pp. 341-352. DOI: 10.1007/978-3-0348-0585-8
12. Banasiak J. On the Application of Substochastic Semigroup Theory to Fragmentation Models with Mass Loss. Journal of Mathematical Analysis and Applications, 2003, vol. 284, no. 1, pp. 9-30. DOI: 10.1016/S0022-247X(03)00154-9
13. Banasiak J., Lachowicz M., Moszynski M. Chaotic Behavior of Semigroups Related to the Process of Gene Amplification-Deamplification with Cell Proliferation. Mathematical Biosciences, 2007, vol. 206, no. 2, pp. 200-215. DOI: 10.1016/j.mbs.2005.08.004
14. Banasiak J., Lamb W. Analytic Fragmentation Semigroups and Continuous Coagulation-Fragmentation Equations with Unbounded Rates. Journal of Mathematical Analysis and Applications, 2012, vol. 391, no. 1, pp. 312-322. DOI: 10.1016/j.jmaa.2012.02.002
15. Dezin, A.A. Differential Operator Equations: A Method of Model Operators in the Theory of Boundary Value Problems. Proceedings of the Steklov Institute of Mathematics, 2000, vol. 229, pp. 1-161. (in Russian)
16. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of 'New Directions in the Mathematical and Computer Sciences', National Universities Commission, Abuja, Nigeria, 2008, pp. 159-232.
17. Shafranov D.E., Adukova N.V. Solvability of the Showalter-Sidorov Problem for Sobolev Type Equations with Operators in the Form of First-Order Polynomials from the Laplace-Beltrami Operator on Differential Forms. Journal of Computation and Engineering Mathematics, 2017, vol. 4, no. 3, pp. 27-34. DOI: 10.14529/jcem170304
18. Moskvicheva P.O., Semenova I.N. The Lyapunov Stability of the Сauchy-Dirichlet Problem for the Generalized Hoff Equation. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 4, pp. 126-131. DOI: 10.14529/mmp140411
19. Keller A.V., Zagrebina S.A. Some Generalizations of the Showalter-Sidorov Problem for Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 5-23. DOI: 10.14529/mmp150201
20. Zamyshlyaeva A.A., Keller A.V., Syropiatov M.B. Stochastic Model of Optimal Dynamic Measurements. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 2, pp. 147-153. DOI: 10.14529/mmp180212
21. Kadchenko S.I., Zakirova G.A. A Numerical Method for Inverse Spectral Problems. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 116-126. DOI: 10.14529/mmp150307
22. Solovyova N.N., Zagrebina S.A. Multipoint Initial-Final Value Problem for Hoff Equation in Quasi-Sobolev Spaces. Journal of Computation and Engineering Mathematics, 2017, vol. 4, no. 2, pp. 73-79 DOI: 10.14529/jcem170208
23. Kadchenko S.I., Soldatova E.A., Zagrebina S.A. Numerical Research of the Barenblatt-Zheltov-Kochina Stochastic Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 2, pp. 117-123. DOI: 10.14529/mmp160211