Semilinear Sobolev Type Mathematical Models

Статья содержит обзор результатов, полученных в научной школе Георгия Анатольевича Свиридюка, в области полулинейных математических моделей соболевского типа. В работе приведены результаты о разрешимости задачи Коши и Шоуолтера - Сидорова для полулинейных уравнений соболевского типа первого, второго и высокого порядков, а также примеры неклассических моделей математической физики, такие, как обобщенная модель нелинейной фильтрации Осколкова, распространения ионно-акустических волн в плазме, распространения волн на мелкой воде, которые исследуются путем редукции к одной из вышеперечисленных абстрактных задач. Методы исследования полулинейных уравнений соболевского типа базируется на теории относительно p-ограниченных операторов для уравнений первого порядка по переменной t и теории относительно полиномиально ограниченных пучков операторов для уравнений второго и высокого порядка по переменной t. В работе применяется метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства, для доказательства теорем существования и единственности и метод Галеркина для построения приближенного решения.

Ключевые слова

уравнение Осколкова; модифицированное уравнение Буссинеска; уравнение ионно-звуковых волн в плазме; полулинейные уравнения соболевского типа; относительно p-ограниченные операторы; относительно полиномиально ограниченные пучки операторов; метод Галеркина; *-слабая сходимость.

Литература

1. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, Walter De Gruyter, 2011. DOI: 10.1515/9783110255294
2. Barenblatt G.I., Zheltov Yu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissurized Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, no. 5, pp. 1286-1303.
3. Bychkov E.V. Analytical Study of the Mathematical Model of Wave Propagation in Shallow Water by the Galerkin Method. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 1, pp. 26-38. DOI: 10.14529/mmp210102
4. Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures. Zeitschrift f"ur angewandte mathematik und physik, 1968, vol. 19, pp. 614-627. DOI: 10.1007/BF01594969
5. Clarkson P.A., Leveque R.J., Saxton R., Solitary Wave Interactions in Elastic Rods. Studies in Applied Mathematics, 1986, vol. 75, no. 1, pp. 95-122. DOI: 10.1002/sapm198675295
6. Cristiansen P.L., Muto V., Lomdahl P.S. On a Toda Lattice Model with a Transversal Degree of Freedom. Nonlinearity, 1991, vol. 4, no. 2, pp. 477-501. DOI: 10.1088/0951-7715/4/2/012
7. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, 2003.
8. Favini A., Sviridyuk G., Manakova N. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of ``Noises''. Abstract and Applied Analysis, 2015, vol. 2015, article ID: 697410.
9. Favini A., Sviridyuk G., Sagadeeva M. 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
10. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, 1999.
11. 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, article ID: 128.
12. Hassard B.D. Theory and Application of Hopf Bifurcation. Cambridge University Press, Camdribge, 1981.
13. Keller A.V., Shestakov A.L., Sviridyuk G.A., Khudyakov Yu.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals. Semigroups of Operators - Theory and Applications. Heidelberg, N.Y., Dordrecht, London, Springer Int. Publ. Switzerland, 2015, pp. 183-195. DOI: 10.1007/978-3-319-12145-1_11
14. Korpusov M.O., Lukyanenko D.V. Instantaneous Blow-up Versus Local Solvability for one Problem of Propagation of Nonlinear Waves in Semiconductors. Journal of Mathematical Analysis and Applications, 2018, vol. 459, no. 1, pp. 159-181. DOI: 10.1016/j.jmaa.2017.10.062
15. Kozhanov A.I. On a Nonlocal Boundary Value Problem with Variable Coefficients for the Heat Equation and the Aller Equation. Differential Equations, 2004, vol. 40, no. 6, pp. 815-826. DOI: 10.1023/B:DIEQ.0000046860.84156.f0
16. Ladyzhenskaya O.A. Matematicheskie voprosy dinamiki vyazkouprugoy neszhimaemoy zhidkosti [Mathematical Problems in the Dynamics of a Viscoelastic Incompressible Fluid]. Moscow, Fizmatgiz, 1961. (in Russian)
17. Lions J.L. Sur quelques methodes de resolution des problemes aux limites non linears. Paris, Dunod, Gauthier Villars, 1969. (in French)
18. Manakova N.A., Gavrilova O.V. About Nonuniqueness of Solutions of the Showalter-Sidorov Problem for One Mathematical Model of Nerve Impulse Spread in Membrane. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 4, pp. 161-168. DOI: 10.14529/mmp180413
19. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Journal of Soviet Mathematics, 1993, vol. 64, no. 1, pp. 724-735. DOI: 10.1007/BF02988478
20. Poincar'e H. Sur l'`equilibre d'une massuide anim'ee d'un mouvement de rotation. Acta Mathematica, 1885, vol. 7, pp. 259-380. DOI: 10.1007/BF02402204 (in French)
21. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, K"oln, Tokyo, VSP, 2002. DOI: 10.1515_9783110900163
22. Sagadeeva M.A., Sviridyuk G.A. The Nonautonomous Linear Oskolkov Model on a Geometrical Graph: The Stability of Solutions and the Optimal Control. Semigroups of Operators - Theory and Applications. Heidelberg, N.Y., Dordrecht, London, Springer Int. Publ. Switzerland, 2015, pp. 257-271. DOI: 10.1007/978-3-319-12145-1_16
23. 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
24. Showalter R.E. Hilbert Space Methods for Partial Differential Equations. Pitman, London, San Francisco, Melbourne, 1977.
25. Shubin Wang, Guowang Chen. Small Amplitude Solutions of the Generalized IMBq Equation. Journal of Mathematical Analysis and Applications, 2002, vol. 274, no. 2, pp. 846-866. DOI: 10.1016/S0022-247X(02)00401-8
26. Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.
27. Sobolev S.L. On a New Problem of Mathematical Physics. Izv. Akad. Nauk SSSR. Ser. Mat., 1954, vol. 18, no. 1, pp. 3-50. (in Russian)
28. Sviridyuk G.A. A Problem of Generalized Boussinesq Filtration Equation. Soviet Math., 1989, vol. 33, no. 2, pp. 62-73.
29. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547
30. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K"oln, Tokyo, VSP, 2003. DOI: 10.1515/9783110915501
31. Sviridyuk G.A., Karamova A.F. On the Phase Space Fold of a Nonclassical Equation. Differential Equation, 2005, vol. 41, no. 10, pp. 1476-1481. DOI: 10.1007/s10625-005-0300-5
32. 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: 10.1023/A:1013919500605
33. Sviridyuk G.A., Kazak V.O. The Phase Space of a Generalized Model of Oskolkov. Siberian Mathematical Journal, 2003, vol. 44, no. 5, pp. 877-882. DOI: 10.1023/A:1026080506657
34. 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
35. Sviridyuk G.A., Manakova N.A. Phase Space of the Cauchy-Dirichlet Problem for the Oskolkov Equation of Nonlinear Filtration. Russian Mathematics, 2003, vol. 47, no. 9, pp. 33-38.
36. 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
37. Sviridyuk G.A., Sukacheva T.G. The Cauchy Problem for a Class of Semilinear Equations of Sobolev Type. Siberian Mathematical Journal, 1990, vol. 31, no. 5, pp. 794-802. DOI: 10.1007/BF00974493
38. Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Spaces of a Class of Linear Higher-Order Sobolev Type Equations. Differential Equations, 2006, vol. 42, no. 2, pp. 269-278. DOI: 10.1134/S0012266106020145
39. Temam R. Navier-Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.Y., Oxford, North Holland Publ. Co., 1979.
40. 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
41. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Second Order Semilinear Sobolev Type Equation. Global and Stochastic Analysis, 2015, vol. 2, no. 2, pp. 159-166.
42. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Sobolev Type Equetion 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
43. Zamyshlyaeva A., Lut A. Inverse Problem for the Sobolev Type Equation of Higher Order. Mathematics, 2021, vol. 9, no. 14, article ID: 1647. DOI: 10.3390/math9141647
44. Zamyshlyaeva A.A., Manakova N.A., Tsyplenkova O.N. Optimal Control in Linear Sobolev Type Mathematical Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 1, pp. 5-27. DOI: 10.14529/mmp200101
45. Zamyshlyaeva A.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 5-16. DOI: 10.14529/mmph160401