On solvability of degenerate linear evolution equations with memory effects

V. E. Fedorov   ; L. V. Borel

The Bulletin of Irkutsk State University. Series Mathematics, Tome 10 (2014), pp. 106-124

Résumé

By the methods of the operators semigroups theory a degenerate linear evolution equation with memory in a Banach space is reduced to a system of two equations. One of them is resolved with respect to the derivative, another has a nilpotent operator at the derivative. A problem with a given history for the first of the equations with memory is brought to the Cauchy problem for stationary equations system in a wider space. It allowed to obtain conditions of the unique solution existence for the problem, including solutions with a greater smoothness, by the methods of the classical operators semigroups theory. Thus unique solvability of the problem with a given history for a degenerate linear evolution equation with memory was researched with using some restrictions for the kernel of the memory integral operator. Besides, an analogous problem with generalized Showalter–Sidorov type condition on the history of the system was studied. General results were used for investigation of an initial boundary value problem for the linearized Oskolkov integro-differential system of equations, descibing the dynamics of the high order Kelvin–Voight fluid.

Détail

Keywords: equation with memory, degenerate evolution equation, operator semigroup, initial boundary value problem, Kelvin–Voight fluid.

Bibliographie

[1] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heideberg–New-York, 1966

[2] Mizokhata S., Theory of Partial Differential Equations, Mir, M., 1977, 504 pp. (in Russian)

[3] Oskolkov A. P., “Initial Boundary Value Problems for Motion Equations of Kelvin–Voight and Oldroyd Fluids”, Proceedings of Steklov Mathematics Institute of USSR Academy of Sciences, 179, 1988, 126–164 (in Russian)

[4] Sidorov N. A., “A Class of Degenerate Differential Equations with Convergence”, Math. Notes, 35:4 (1984), 300–305 | DOI

[5] Stakheeva O. A., “Local Solvability of a Class of Linear Equations with Memory”, Herald of Chelyabinsk State University. Ser. Mathematics, Mechanics, Informatics, 20(158):11 (2009), 70–76 (in Russian)

[6] Falaleev M. V., “Integro-Differential Equations with Fredholm Operator at Highest Derivative in Banach Spaces and Their Applications”, News of Irkutsk State University. Ser. Mathematics, 5:2 (2012), 90–102 (in Russian)

[7] Falaleev M. V., Orlov S. S., “Degenerate Integro-Differential Operators in Banach Spaces and Their Applications”, Russian Math. (Iz. VUZ), 55:10 (2011), 59–69

[8] Falaleev M. V., Orlov S. S., “Degenerate Integro-Differential Equations of Special Form in Banach Spaces and Their Applications”, Herald of South Ural State University, Ser. Mathematical Modeling and Programming, 35(211):6 (2010), 104–109 (in Russian)

[9] Falaleev M. V., Orlov S. S., “Integro-Differential Equations with Degeneration in Banach Spaces and Their Applications in Mathematical Elasticity Theory”, News of Irkutsk State University, 4:1 (2011), 118–134 (in Russian)

[10] Fedorov V. E., Omelchenko O. A., “Inhomogeneous Degenerate Sobolev Type Equations with Delay”, Siberian Mathematical Journal, 53:2 (2012), 335–344 | DOI

[11] Fedorov V. E., Omelchenko O. A., “Linear Equations of the Sobolev Type with Integral Delay Operator”, Russian Math. (Iz. VUZ), 58:1 (2014), 60–69

[12] Fedorov V. E., Stakheeva O. A., “On Solvability of Linear Sobolev Type Equations with Memory Effect”, Nonclassical Mathematical Physics Equations, Sobolev Institute of Mathematics of SB RAS, Novosibirsk, 2010, 245–261 (in Russian)

[13] C. Giorgi, A. Marzocchi, “Asymptotic behavior of a semilinear problem in heat conduction with memory”, Nonlinear Differ. Equ. Appl., 5 (1998), 333–354 | DOI

[14] M. E. Gurtin, A. C. Pipkin, “A general theory of heat conduction with finite wave speeds”, Arch. Rational Mech. Anal., 31 (1968), 113–126 | DOI

[15] N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev, Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 2002, 548 pp.

[16] S. Gatti, M. Grasselli, V. Pata, M. Squassina, “Robust exponential attractors for a family of nonconserved phase-field systems with memory”, Discrete and Continuous Dynamical Systems, 12:5 (2005), 1019–1029 | DOI

[17] R. E. Showalter, “Nonlinear degenerate evolution equations and partial differential equations of mixed type”, SIAM J. Math. Anal., 6:1 (1975), 25–42 | DOI

[18] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston, 2003, 216+vii pp.