## List of issues > Series «Mathematics». 2014. Vol. 10

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On Solvability of Degenerate Linear Evolution Equations with Memory Effects

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 problemь 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.

1. Falaleev M. V. Integro-differentsial’nye uravneniya s fredgol’movym operatorom pri starshey proizvodnoy v banakhovykh prostranstvakh i ikh prilozheniya (in Russian) [Integro-Differential Equations with Fredholm Operator at Highest Derivative in Banach Spaces and Their Applications]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika [News of Irkutsk State University, Ser. Mathematics], 2012, vol. 5, no. 2, pp. 90-102.

2. Falaleev M. V., Orlov S. S. Degenerate Integro-Differential Operators in Banach Spaces and Their Applications. Russian Math. (Iz. VUZ), 2011, vol. 55, no. 10, pp. 59-69.

3. Falaleev M. V., Orlov S. S. Vyrozhdennye integro-differentsial’nye uravneniya spetsial’nogo vida v banakhovykh prostranstvakh i ikh prilozheniya (in Russian) [Degenerate Integro-Differential Equations of Special Form in Banach Spaces and Their Applications]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Ser. Matematicheskoe modelirovanie i programmirovanie [Herald of South Ural State University, Ser. Mathematical Modeling and Programming], 2010, issue 6, no. 35 (211), pp. 104-109.

4. Falaleev M. V., Orlov S. S. Integro-differentsial’nye uravneniya s vyrozhdeniem v banakhovykh prostranstvakh i ikh prilozheniya v matematicheskoy teorii uprugosti (in Russian) [Integro-Differential Equations with Degeneration in Banach Spaces and Their Applications in Mathematical Elasticity Theory]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika [News of Irkutsk State University], 2011, vol. 4, no. 1, pp. 118-134.

5. Fedorov V. E., Omelchenko O. A. Inhomogeneous Degenerate Sobolev Type Equations with Delay. Siberian Mathematical Journal, 2012, vol. 53, no. 2, pp. 418-429.

6. Fedorov V. E., Omelchenko O. A. Linear Equations of the Sobolev Type with Integral Delay Operator. Russian Math. (Iz. VUZ), 2014, vol. 58, no. 1, pp. 60-69.

7. Fedorov V. E., Stakheeva O. A. O razreshimosti lineynykh uravneniy sobolevskogo tipa s effektom pamyati (in Russian) [On Solvability of Linear Sobolev Type Equations with Memory Effect]. Neklassicheskie uravneniya matematicheskoy fiziki[Nonclassical Mathematical Physics Equations], Novosibirsk, Sobolev Institute of Mathematics of SB RAS, 2010, pp. 245-261.

8. Gatti S., Grasselli M., Pata V., Squassina M. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete and Continuous Dynamical Systems, 2005, vol. 12, no. 5, pp. 1019-1029.

9. Giorgi C., Marzocchi A. Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differ. Equ. Appl., 1998, vol. 5, pp. 333-354.

10. Gurtin M. E., Pipkin A. C. A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal., 1968, vol. 31, pp. 113-126.

11. Kato T. Perturbation Theory for Linear Operators, Berlin–Heidelberg–New York, Springer-Verlag, 1966.

12. Mizokhata S. Teoriya uravneniy s chastnymi proizvodnymi (in Russian) [Theory of Partial Differential Equations], Moscow, Mir, 1977, 504 p.

13. Oskolkov A. P. Nachal’no-kraevye zadachi dlya uravneniy dvizheniya zhidkostey Kelvina – Voighta i zhidkostey Oldroyda (in Russian) [Initial Boundary Value Problems for Motion Equations of Kelvin–Voight and Oldroyd Fluids]. TrudyMat. Instituta AN SSSR [Proceedings of Steklov Mathematics Institute of USSR Academy of Sciences], 1988, vol. 179, pp. 126-164.

14. Showalter R. E. Nonlinear degenerate evolution equations and partial differentialequations of mixed type. SIAM J. Math. Anal., 1975, vol. 6, no. 1, pp. 25-42.

15. Sidorov N. A. A Class of Degenerate Differential Equations with Convergence. Mathematical Notes, 1984, vol. 35, no. 4, p. 300-305.

16. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov – Schmidt Methods in Nonlinear Analysis and Applications, Dordrecht, Kluwer Acad. Publ., 2002, 548 p.

17. Stakheeva O. A. Local’naya razreshimost’ odnogo klassa lineynykh uravneniy s pamyat’yu (in Russian) [Local Solvability of a Class of Linear Equations with Memory]. Vestnik Chelyabinskogo gosudarstvennogo universiteta. Ser.Matematika. Mekhanika. Informatika [Herald of Chelyabinsk State University. Ser. Mathematics, Mechanics, Informatics], 2009, issue 11, no. 20 (158), pp. 70-76.

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