Semilinear Sobolev type mathematical models

Semilinear Sobolev type mathematical models

Автор: Zamyshlyaeva A.A., Bychkov E.V.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp

Рубрика: Обзорные статьи

Статья в выпуске: 1 т.15, 2022 года.

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The article contains a review of the results obtained in the scientific school of Georgy Sviridyuk in the field of semilinear Sobolev type mathematical models. The paper presents results on solvability of the Cauchy and Showalter-Sidorov problems for semilinear Sobolev type equations of the first, the second and higher orders, as well as examples of non-classical models of mathematical physics, such as the generalized Oskolkov model of nonlinear filtering, propagation of ion-acoustic waves in plasma, propagation waves in shallow water, which are studied by reduction to one of the above abstract problems. Methods for studying the semilinear Sobolev type equations are based on the theory of relatively -bounded operators for equations of the first order and the theory of relatively polynomially bounded operator pencils for equations of the second and higher orders in the variable . The paper uses the phase space method, which consists in reducing a singular equation to a regular one defined on some subspace of the original space, to prove existence and uniqueness theorems, and the Galerkin method to construct an approximate solution.

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Oskolkov equation, equation of ion-acoustic waves in plasma, modified boussinesq equation, semilinear sobolev type equation, relatively -bounded operators, relatively polynomially bounded operator pencils, galerkin method, *-weak convergence

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Короткий адрес: https://sciup.org/147237428

IDR: 147237428

Список литературы Semilinear Sobolev type mathematical models

  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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.
  • 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
  • Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, 1999.
  • 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.
  • Hassard B.D. Theory and Application of Hopf Bifurcation. Cambridge University Press, Camdribge, 1981.
  • 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
  • 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
  • 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
  • Ladyzhenskaya O.A. Matematicheskie voprosy dinamiki vyazkouprugoy neszhimaemoy zhidkosti [Mathematical Problems in the Dynamics of a Viscoelastic Incompressible Fluid]. Moscow, Fizmatgiz, 1961. (in Russian)
  • Lions J.L. Sur quelques methodes de resolution des problemes aux limites non linears. Paris, Dunod, Gauthier Villars, 1969. (in French)
  • 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
  • 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
  • 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)
  • Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, K¨oln, Tokyo, VSP, 2002. DOI: 10.1515_9783110900163
  • 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
  • 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
  • Showalter R.E. Hilbert Space Methods for Partial Differential Equations. Pitman, London, San Francisco, Melbourne, 1977.
  • ShubinWang, 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
  • Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov–Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.
  • 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)
  • Sviridyuk G.A. A Problem of Generalized Boussinesq Filtration Equation. Soviet Math., 1989, vol. 33, no. 2, pp. 62–73.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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
  • 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
  • Temam R. Navier–Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.Y., Oxford, North Holland Publ. Co., 1979.
  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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
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