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Exponential Stability of Semigroups Generated by Volterra Integro-Differential Equations with Singular Kernels

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Abstract

In a separable Hilbert space, we study abstract linear inhomogeneous second-order Volterra integro-differential equations on the positive half-line with operator coefficients and with kernels that have integrable singularities. The equations under consideration are operator models of problems arising in the theory of viscoelasticity. We give a method for reducing the initial value problem for an equation of this class to the Cauchy problem for a linear differential equation in an extended function space. Sufficient conditions on the kernels of the integral operators in the integro-differential equation are found for the corresponding linear homogeneous differential equation in the extended space to have an exponentially stable contraction semigroup. Estimates are obtained for the solutions of the Cauchy problem for the linear equation in the extended space. By way of example, the results are applied to the case of fractional exponential kernels (Rabotnov functions) of integral operators.

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Funding

This work was carried out with partial financial support from the Ministry of Education and Science of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2019-1621 (Theorem 3) and with partial financial support from the Russian Foundation for Basic Research (Theorem 4), project no. 20-01-00288 A).

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    V. V. Vlasov & N. A. Rautian

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    V. V. Vlasov & N. A. Rautian

Authors
  1. V. V. Vlasov
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  2. N. A. Rautian
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Correspondence to V. V. Vlasov or N. A. Rautian.

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Translated by V. Potapchouck

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Vlasov, V.V., Rautian, N.A. Exponential Stability of Semigroups Generated by Volterra Integro-Differential Equations with Singular Kernels. Diff Equat 57, 1402–1407 (2021). https://doi.org/10.1134/S0012266121100141

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  • DOI: https://doi.org/10.1134/S0012266121100141

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