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Vitali Vougalter

Publications and source records attributed to Vitali Vougalter.

At least 19 recordsLinked to original sources
arXiv2026

Solvability in the sense of sequences for certain non-Fredholm operators with a drift and Laplace and bi-Laplace operators

We study the solvability of some linear nonhomogeneous elliptic problems and establish that under certain technical assumptions the convergence in $L^2$ of their right-hand sides yields the existence and the convergence in $H^4$ of the solutions. The equations contain fourth order differential operators with or without the Fredholm property, in particular the second and the fourth derivative operators, on the whole real line or on a finite interval with periodic boundary conditions. We establish that the transport term involved in these problems provides the regularization of the solutions.

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arXiv2026

Existence of stationary solutions for some systems of integro-differential equations with Laplace and bi-Laplace operators

The article is devoted to the solvability of a system of integro-differential equations in the case of the difference of the standard Laplacian and the bi-Laplacian in the diffusion terms. The proof of the existence of solutions is based on a fixed point technique. We use the solvability conditions for the elliptic operators without the Fredholm property in unbounded domains.

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arXiv2026

Solvability of a class of integro-differential equations with Laplace and bi-Laplace operators

The work deals with the studies of the existence of solutions of an integro-differential equation in the situation of the difference of the standard Laplacian and the bi-Laplacian in the diffusion term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains.

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arXiv2026

On the well-posedness of a certain model with two kernels appearing in the mathematical biology

The work is devoted to establishing the global well-posedness in $W^{(1,2),2}(R\times R^{+})$ of the integro-differential problem involving the two nonlocal terms describing the diffusion and the production in the biological system in the presence of the transport term. Such model is relevant to the cell population dynamics in the Mathematical Biology. The proof is based on a fixed point technique.

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arXiv2025

Solvability conditions for some non-Fredholm operators with shifted arguments

In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is shown that under the reasonable technical conditions the convergence in $L^{2}(R)$ of the source terms implies the existence and the convergence in $H^{2}(R)$ of the solutions. The second part of the work deals with the solvability in the sense of sequences in $H^{2}(R)$ of the integro-differential equation in which one of the terms has the argument shifted by a constant. It is demonstrated that under the appropriate auxiliary assumptions the convergence in $L^{1}(R)$ of the integral kernels yields the existence and the convergence in $H^{2}(R)$ of the solutions. Both equations considered involve the second order differential operator with or without the Fredholm property depending on the value of the constant by which the argument gets translated.

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arXiv2025

The preservation of nonnegativity of solutions of a parabolic system with the cubed Laplacian

The article is devoted to the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations containing the cubed Laplacian. Such necessary condition is extremely important for the applied analysis community since it imposes the necessary form of the system of equations that must be studied mathematically.

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arXiv2025

Solvability of some integro-differential equations with the bi-Laplacian and transport

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H^4 spaces. Our argument is based on the fixed point technique when the elliptic equations contain fourth order differential operators with and without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L^1 of the integral kernels yields the existence and convergence in H^4 of the solutions.

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arXiv2025

On the existence of stationary solutions for certain systems of integro-differential equations with the double scale anomalous diffusion

The work deals with establishing the solvability of a system of integro-differential equations in the situation of the double scale anomalous diffusion. Each equation of such system involves the sum of the two negative Laplace operators raised to two distinct fractional powers in the space of three dimensions. The proof of the existence of solutions is based on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains.

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arXiv2025

On the well-posedness of some model with the cubed Laplacian arising in the Mathematical Biology

In the article we establish the global well-posedness in W^{1,(6,2)}(R \times R+) of the integro-differential equation containing the cube of the one dimensional Laplacian and the transport term. Our proof relies on a fixed point technique. Furthermore, we formulate the condition leading to the existence of the nontrivial solution for our problem under the consideration. This problem is relevant to the cell population dynamics in the Mathematical Biology.

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arXiv2025

On the well-posedness of a certain model with the bi-Laplacian appearing in the Mathematical Biology

The work is devoted to the global well-posedness in W^{1, (4, 2)}(R\times R^{+}) of the integro-differential problem involving the square of the one dimensional Laplace operator along with the drift term. Our proof is based on a fixed point technique. Moreover, we provide the assumption leading to the existence of the nontrivial solution for the problem under the consideration. Such equation is relevant to the cell population dynamics in the Mathematical Biology.

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arXiv2024

Existence of stationary solutions for some integro-differential equations with the double scale anomalous diffusion

The paper is devoted to the investigation of the solvability of an integro-differential equation in the case of the double scale anomalous diffusion with a sum of two negative Laplacians in different fractional powers in R^3. The proof of the existence of solutions relies on a fixed point technique. Solvability conditions for the elliptic operators without the Fredholm property in unbounded domains are used.

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arXiv2024

Existence of solutions for some systems of superdiffusive integro-differential equations in population dynamics depending on the natality and mortality rates

We prove the existence of stationary solutions for some systems of reaction-diffusion type equations with superdiffusion in the corresponding H^2 spaces. Our method is based on the fixed point theorem when the elliptic problems contain first order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates contained in the equations of our systems.

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arXiv2024

Solvability of some integro-differential equations with the logarithmic Laplacian

We address the existence in the sense of sequences of solutions for a certain integro-differential type problem involving the logarithmic Laplacian. The argument is based on the fixed point technique when such equation contains the operator without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L^1(R^d) of the integral kernels yields the existence and convergence in L^2(R^d) of the solutions.

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arXiv2024

On the existence of solutions for a class of systems of integro-differential equations with the logarithmic Laplacian and drift

In this article, we consider a system of integro-differential equations in L^2(R, R^N), which contains the logarithmic Laplacian in the presence of transport terms. The linear operators associated with the system satisfy the Fredholm property. By virtue of a fixed point technique, we demonstrate the existence of solutions. We emphasize that the discussion is more complicated than that of the scalar situation as there are more cumbersome technicalities to overcome.

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