Correct solvability and exponential stability for solutions of Volterra integro-differential equations

N. A. Rautian

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 62-66.

Résumé

Abstract integro-differential equations that are operator models of viscoelasticity problems are studied. The kernels of the integral operators can be specified as sums of decreasing exponentials or sums of Rabotnov functions with positive coefficients, which are widely used in viscoelasticity theory. A method is described whereby the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space is reduced to the Cauchy problem for a first-order differential equation. Exponential stability of solutions is established under known assumptions on the kernels of the integral operators. The results are used to establish the correct solvability of the original initial value problem for a Volterra integro-differential equation with corresponding solution estimates.

Détail

Keywords: Volterra integro-differential equations, linear differential equations in Hilbert spaces, exponential stability.

Bibliographie

[1] Ilyushin A.A., Pobedrya B.E., Osnovy matematicheskoi teorii termovyazkouprugosti, Nauka, M., 1970, 280 pp. | MR

[2] Christensen R.M., Theory of viscoelasticity. An introduction, Academic Press, N.Y.–L., 1971, 364 pp.

[3] Kopachevsky N.D., Krein S.G., Operator Approach to Linear Problems of Hydrodynamics, v. 2, Operator Theory: Advances and Applications, 146, Nonself-adjoint Problems for Viscous Fluids, Birkhauser Verlag, Basel/Switzerland, 2003, 444 pp. | MR | Zbl

[4] Munoz Rivera J.E., “Asymptotic behaviour in linear viscoelasticity”, Quart. Appl. Math., 52 (1994), 629–648 | MR | Zbl

[5] Amendola G., Fabrizio M., Golden J.M., Thermodynamics of Materials with memory. Theory and applications, Springer, N.Y.–Dordrecht–Heidelberg–L., 2012, 576 pp. | MR | Zbl

[6] Lokshin A.A., Suvorova Yu.V., Matematicheskaya teoriya rasprostraneniya voln v sredakh s pamyatyu, Izd-vo MGU, M., 1982, 152 pp. | MR

[7] Gurtin M.E., Pipkin A.C., “General theory of heat conduction with finite wave speed”, Arch. Rat. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

[8] Lykov A.V., Teplomassoobmen, Spravochnik, 2-e izd., pererab. i dop., Energiya, M., 1978, 480 pp.

[9] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[10] Rabotnov Yu.N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977, 384 pp. | MR

[11] Krein S.G., Lineinye differentsialnye uravneniya v banakhovykh prostranstvakh, Nauka, M., 1967, 464 pp.

[12] Engel K.J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, N.Y., 2000, 586 pp. | MR | Zbl

[13] Vlasov V.V., Rautian N.A., “Correct solvability and representation of solutions of volterra integrodifferential equations with fractional exponential kernels”, Doklady Mathematics, 100:2 (2019), 467–471 | DOI | MR | Zbl

[14] Vlasov V.V., Rautian N.A., “A Study of Operator Models Arising in Problems of Hereditary Mechanics”, Journal of Mathematical Sciences (N.Y.), 244:2 (2020), 170–182 | DOI | MR | Zbl

[15] Skubachevskii A.L., “Boundary-value problems for elliptic functional-differential equations and their applications”, Russian Mathematical Surveys, 71:5 (2016), 801–906 | DOI | MR | Zbl