Well-Posedness and Spectral Analysis of Integrodifferential Equations Arising in Viscoelasticity Theory

Власов, В. В.; Раутиан, Н. А.

doi:10.1007/s10958-018-3943-5

Cited by 16 publications

(2 citation statements)

References 18 publications

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“…It should be noted that non-differential operators contained in the studied equations might be quite diverse. For instance, they might be integrodifferential operators (see, e.g., [10][11][12][13][14][15][16] and references therein), operators of contractions and extensions of the independent variables (see, e.g., [17][18][19][20][21] and references therein), or others (see, e.g., [22,23] and references therein). In general, those operators are bounded (unlike differential ones), but due to their nonlocal nature, they cannot be treated as subordinate terms or small perturbations: their presence cause qualitatively new properties of the solutions.…”

Section: Differential-difference Equationsmentioning

confidence: 99%

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

Muravnik

2023

Mathematics

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We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity.

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Section: Differential-difference Equationsmentioning

confidence: 99%

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

Muravnik

2023

Mathematics

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“…Such equations form a special (though quite important) subclass of the class of functional differential equations, i. e., equations with arbitrary non-differential operators acting (apart from differential ones) on the desired function. Those non-differential operators might be integrodifferential ones (see, e. g., [2][3][4][5][6][7][8] and references therein), operators of contractions and extensions of the independent variables (see, e.g., [9][10][11][12][13] and references therein), or others (see, e.g., [14,15] and references therein). Although those operators are, in general, bounded (unlike differential ones), they cannot be treated as small perturbations or subordinate terms of the equation: they are nonlocal terms, and, as we see in various investigations, the presence of such terms implies the presence of qualitatively new properties of the solutions.…”

Section: Introductionmentioning

confidence: 99%

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

Muravnik,

Rossovskii

2024

Mathematics

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We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well.

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Well-Posedness of Volterra Integro-Differential Equations with Fractional Exponential Kernels

Раутиан

2020

Springer Proceedings in Mathematics &Amp; Statistics

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Well-Posedness and Spectral Analysis of Integrodifferential Equations Arising in Viscoelasticity Theory

Cited by 16 publications

References 18 publications

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

Cauchy Problem with Summable Initial-Value Functions for Parabolic Equations with Translated Potentials

Well-Posedness of Volterra Integro-Differential Equations with Fractional Exponential Kernels

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