Peculiarities of the Phase Manifold of One Sobolev-Type Equation

The problem of the non-uniqueness of the solution of the Cauchy problem is one of the fundamental challenges in the theory of differential equations, demonstrating that even with the apparent completeness of the formulation, the further evolution of the system can be unpredictable. In the case of degenerate nonlinear models of mathematical physics, it is also possible that the solution of the initial problem may not be unique and that Whitney assemblies may appear in phase manifolds. The issues of non-uniqueness of solutions to initial boundary value problems for equations related to semilinear Sobolev type equations and the connection between the non-uniqueness of solutions to the problem and the existence of the Whitney equation of assemblies and folds in the phase manifold are investigated. The mathematical Hoff model is studied and the conditions imposed on the parameters of the equation are revealed, under which the phase manifold has singularities, and there are several solutions to the Showalter-Sidorov problem in the case when the dimension of the kernel of the operator with a time derivative is arbitrary.

Keywords

Sobolev type equations; Whitney assemblies; phase space method.

References

1. Zakharov S.V. Singular Asymptotics in the Cauchy Problem for a Parabolic Equation with a Small Parameter. Proceedings of the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2015, vol. 21, no. 1, pp. 97-104.
2. R. de Sa Teles Generalized Semiflows for a Plate Model with Presumed Nonuniqueness of Solution. Rocky Mountain Journal of Mathematics, 2019, vol. 49, no. 6, pp. 2047-2061. DOI: 10.1216/RMJ-2019-49-6-2047
3. Bokarieva T.A., Sviridiuk G.A. Whitney Folds in Phase Spaces of Some Semilinear Sobolev-Type Equations. Mathematical Notes, 1994, vol. 55, no. 3, pp. 237-242. DOI:10.1007/BF02110776
4. 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. Serirs: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 5-16. DOI: 10.14529/mmph160401
5. Manakova N.A., Gavrilova O.V., Perevozhikova K.V. Semilinear Models of Sobolev Type. Non-Uniqueness of Solution to the Showalter-Sidorov Problem. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2022, vol. 15, no. 1, pp. 84-100. DOI: 10.14529/mmp220105
6. Hoff N.J. Creep Buckling. Journal of the Aeronautical Science, 1956, no. 7, pp. 1-20.
7. Zagrebina S.A. The Multipoint Initial-Finish Problem for Hoff Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 11, pp. 4-12.
8. Sidorov N.A., Romanova O.A. On the Application of Certain Results of Bifurcation Theory to the Solution of Degenerate Differential Equations. Differential Equations, 1983, vol. 19, no. 9, pp. 1516-1526.
9. Nikolaeva N.G., Gavrilova O.V., Manakova N.A. Investigation of the Uniqueness Solution of the Showalter-Sidorov Problem for the Mathematical Hoff Model. Phase Space Morphology. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2024, vol. 17, no. 1, pp. 49-63. DOI: 10.14529/mmp240105
10. 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. 1-2, pp. 262-266. DOI: 10.4213/mzm347
11. Sviridyuk G.A., Trineeva I.K. A Whitney Fold in the Phase Space of the Hoff Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2005, vol. 49, no. 10, pp. 49-55.
12. Sviridyuk G.A., Zagrebina S.A. The Showalter-Sidorov Problem as a Phenomena of the Sobolev-Type Equations. Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, no. 1, pp. 104-125.
13. Sviridyuk G.А., Sukacheva T.G. The Phase Space of a Class of Operator Equations of Sobolev Type. Differential Equations, 1990, vol. 26, no. 2, pp. 250-258.
14. Gil'mutdinova A.F. On the Non-Uniqueness of Solutions of Showalter-Sidorov Problem for One Plotnikov Model. Vestnik of Samara State University, 2007, no. 9/1, pp. 85-90. (in Russian)
15. Sviridyuk G.A., Karamova A.F. On the Phase Space Fold of a Nonclassical Equation. Differential Equations, 2005, vol. 41, no. 10, pp. 1476-1481. DOI: 10.1007/s10625-005-0300-5
16. Arnold V.I. Catastrophe Theory. Moscow, Nauka, 1990. (in Russian)
17. Gilmore R. Catastrophe Theory for Scientists and Engineers. New York, Wiley, 1981.
18. Thom R. Stabilite Structurelle et Morphogenese. Paris, Benjamin, 1972. (in French)
19. Manakova N.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Phase Space of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31-51. DOI: 10.14529/mmph160304