Geodesic
Semigroups of operators generated by integro-differential equations with kernels representable by Stieltjes integrals
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 507-525 
V. V. Vlasov; N. A. Rautian. Semigroups of operators generated by integro-differential equations with kernels representable by Stieltjes integrals. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 507-525. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a6/
@article{CMFD_2021_67_3_a6,
     author = {V. V. Vlasov and N. A. Rautian},
     title = {Semigroups of operators generated by integro-differential equations with kernels representable by {Stieltjes} integrals},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {507--525},
     year = {2021},
     volume = {67},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a6/}
}
TY  - JOUR
AU  - V. V. Vlasov
AU  - N. A. Rautian
TI  - Semigroups of operators generated by integro-differential equations with kernels representable by Stieltjes integrals
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2021
SP  - 507
EP  - 525
VL  - 67
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a6/
LA  - ru
ID  - CMFD_2021_67_3_a6
ER  - 
%0 Journal Article
%A V. V. Vlasov
%A N. A. Rautian
%T Semigroups of operators generated by integro-differential equations with kernels representable by Stieltjes integrals
%J Contemporary Mathematics. Fundamental Directions
%D 2021
%P 507-525
%V 67
%N 3
%U http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a6/
%G ru
%F CMFD_2021_67_3_a6

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Abstract Volterra integro-differential equations with kernels of integral operators representable by Stieltjes integrals are investigated. The presented results are based on the approach related to the study of one-parameter semigroups for linear evolution equations. We present the method of reduction of the original initial-value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space. The existence of the contractive $C_0$-semigroup is proved. An estimate for the exponential decay of the semigroup is obtained.

[1] Bellman R., Teoriya ustoichivosti reshenii differentsialnykh uravnenii, IL, M., 1954

[2] Vlasov V. V., Rautian N. A., Spektralnyi analiz funktsionalno-differentsialnykh uravnenii, MAKS Press, M., 2016

[3] Ilyushin A. A., Pobedrya B. E., Osnovy matematicheskoi teorii termovyazkouprugosti, Nauka, M., 1970

[4] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[5] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[6] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1989

[7] Lokshin A. A., Suvorova Yu. V., Matematicheskaya teoriya rasprostraneniya voln v sredakh s pamyatyu, MGU, M., 1982

[8] Lykov A. V., “Nekotorye problemnye voprosy teorii teplomassoperenosa”, Problemy teplo- i massoperenosa, Nauka i tekhnika, Minsk, 1976, 9–82

[9] Rabotnov Yu. N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977

[10] Sanches Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984

[11] Smirnov V. I., Kurs vysshei matematiki, v. 5, Nauka, M., 1974

[12] Amendola G., Fabrizio M., Golden J. M., Thermodynamics of materials with memory. Theory and applications, Springer, New-York–Dordrecht–Heidelberg–London, 2012 | Zbl

[13] Christensen R. M., Theory of viscoelasticity. An introduction, Academic Press, New York–London, 1971

[14] Dafermos C. M., “Asymptotic stability in viscoelasticity”, Arch. Ration. Mech. Anal., 37 (1970), 297–308 | DOI | Zbl

[15] Engel K.-J., Nagel R., One-parameter semigroup for linear evolution equations, Springer, New York, 1999

[16] Eremenko A., Ivanov S., “Spectra of the Gurtin–Pipkin type equations”, SIAM J. Math. Anal., 43 (2011), 2296–2306 | DOI | Zbl

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

[18] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Nonself-adjoint problems for viscous fluids, Birkhäuser, Basel, 2003 | Zbl

[19] Miller R. K., “An integrodifferencial equation for rigid heat conductors with memory”, J. Math. Anal., 66 (1978), 313–332 | DOI | Zbl

[20] Munoz Rivera J. E., Naso M. G., Vegni F. M., “Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory”, J. Math. Anal. Appl., 286 (2003), 692–704 | DOI | Zbl

[21] Pata V., “Stability and exponential stability in linear viscoelasticity”, Milan J. Math., 77 (2009), 333–360 | DOI | Zbl

[22] Rautian N. A., “Semigroups generated by Volterra integro-differential equations”, Differ. Equ., 56:9 (2020), 1193–1211 | DOI | Zbl

[23] Tretter C., Spectral theory of block operator matrices and applications, Imperial College Press, London, 2008 | Zbl

[24] Vlasov V. V., Rautian N. A., “Spectral analysis of integrodifferential equations in Hilbert spaces”, J. Math. Sci. (N.Y.)., 239:5 (2019), 771–787 | DOI | Zbl

[25] Vlasov V. V., Rautian N. A., “On Volterra integro-differential equations with kernels representable by Stieltjes integrals”, Differ. Equ., 57:4 (2021), 517–532 | DOI | Zbl