Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms

Antont︠s︡ev, S. N.; Khompysh, Kh.

doi:10.1016/j.jmaa.2017.06.056

Cited by 17 publications

(12 citation statements)

References 13 publications

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“…Жоғарыдағы (1) түрдегі псевдопараболалық теңдеулер ньютондық емес сұйықтардың, дәлірек айтқанда Кельвин-Фойгт сұйығының қозғалысының бір өлшемді жағдайын сипаттайды [2]. Мұндай р-Лапласианды Кельвин-Фойгт теңдеуі үшін әр түрлі қойылымдардағы сызықты есептердің шешімділігі мен шешімнің сапалық қасиеттерін зерттеу Антоцев пен Хомпыштың [3]- [4] жұмыстарында зерттелінген. Атап айтқанда, сызықты емес Кельвин-Фойгт теңдеуі үшін бастапқышеттік есептің шешімінің глобалды әрі локальды бар болу және жалғыздығы, сонымен қатар шешімнің ақырлы уақыттағы қирауы сынды сапалық қасиеттері зерттелінген.…”

Section: есептің қойылымы және кіріспеunclassified

ASYMTOTIC PROPERTIES OF THE GENERALIZED SOLUTION OF NONLINEAR PROBLEM FOR THE PSEUDOPARABOLIC EQUATION WITH p,q-LAPLACIAN STRUCTURE

Nugmanova¹

2020

Bullet. Ser. of Phys. & Math. Sc.

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The development of modern science and technology requires accurate and comprehensive mathematical modeling of physical and hydrodynamic processes of continuous media. For example, experimental studies show that the presence of even small amounts of polymeric substances in a viscous medium can significantly affect to motion of fluid. A comparison of the physical characteristics of water and weak aqueous solutions of polymers showed that at almost the same density and viscosity, these fluids sharply differ in their relaxation properties. In recent years, the investigation of solvability and qualitative properties such as blow up and extinction of a solution in a finite time, large time behavior of a solution of the problems in various statements for pseudo-parabolic equations has been rapidly developing. This is confirmed by publications have been publishing in world ranking scientific journals. This paper is devoted to the study of a mathematical model of the one-dimensional motion of a Kelvin-Voigt fluid with complex rheological properties, which describes by a nonlinear pseudo-parabolic equation (Sobolev type equation) with p-Laplacian. The study of pseudo-parabolic equations was first started by Sobolev in [1] for the linear version. In this work the asymptotic behavior of solutions of pseudo-parabolic equations at large times, in clearly, the properties of exponential and power-law decay of a solution are proved.

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Section: есептің қойылымы және кіріспеunclassified

ASYMTOTIC PROPERTIES OF THE GENERALIZED SOLUTION OF NONLINEAR PROBLEM FOR THE PSEUDOPARABOLIC EQUATION WITH p,q-LAPLACIAN STRUCTURE

Nugmanova¹

2020

Bullet. Ser. of Phys. & Math. Sc.

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“…The systems of nonlinear equations of the Sobolev type (Kelvin–Voigt equations) describe flows of viscoelastic fluids. Investigations of the mathematical correctness of such equations are devoted to the works 14–16,32–37 …”

Section: Introductionmentioning

confidence: 99%

“…The study of problems for the pseudo‐parabolic type began in the late 1970s. A large number of works are devoted to the study of nonlinear equations of pseudoparabolic type 2–37 . The modeling of physical processes leading to equations of the Sobolev type and, in particular, of the pseudoparabolic type is devoted to the works 2,3,5,6,14–19,23 …”

Section: Introductionmentioning

confidence: 99%

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Antont︠s︡ev

Aitzhanov

Zhanuzakova

2022

Math Methods in App Sciences

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An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained. KEYWORDS asymptotic behavior of the solution, blow-up of the solution, Galerkin method, nonlinear boundary conditions, pseudoparabolic equations, the existence of a solution, uniqueness of the solution MSC CLASSIFICATION

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“…When F ( x , t ) = f ( t ) g ( x , t ) is given and

p = 2, γ = 0

, the various direct problems for classical linear and nonlinear Kelvin–Voigt equations have been studied by Oskolkov 8,11–13 . The nonlinear direct problems for generalized Kelvin–Voigt equations () and () perturbed by isotropic or anisotropic diffusion, relaxation, and damping for homogeneous and also nonhomogeneous fluids, have been investigated in previous studies, 14–20 where the global and local existence and uniqueness, and the quality properties of weak solutions are established.…”

Section: Introductionmentioning

confidence: 99%

“…[8][9][10] When F(x, t) = f (t)g(x, t) is given and p = 2, 𝛾 = 0, the various direct problems for classical linear and nonlinear Kelvin-Voigt equations have been studied by Oskolkov. 8,[11][12][13] The nonlinear direct problems for generalized Kelvin-Voigt equations (1.1) and (1.2) perturbed by isotropic or anisotropic diffusion, relaxation, and damping for homogeneous and also nonhomogeneous fluids, have been investigated in previous studies, [14][15][16][17][18][19][20] where the global and local existence and uniqueness, and the quality properties of weak solutions are established. Some inverse problems for linear and nonlinear classical Kelvin-Voigt equations with integral overdetermination condition, which consist of determining a right hand side depending on time or space variable, have been studied by Abylkairov and Khompysh in Abylkairov and Khompysh 21 and Khompysh, 4 where the unique solvability is established.…”

Section: Introductionmentioning

confidence: 99%

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

Khompysh

Kenzhebai

2021

Math Methods in App Sciences

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In this paper, we consider an inverse problem of finding a coefficient of right hand side of the following system of Kelvin–Voigt equations perturbed by an isotropic diffusion and damping terms vt+∇π−ϰΔvt−νdiv|D(v)|p−2D(v)=γ|v|m−2v+f(t)g(x,t), divv(x,t)=0. The damping term γ|v|m − 2v in the momentum equation realizes an absorbtion (sink) if γ ≤ 0, and a source if γ > 0. We show how the exponents p, m, the coefficients ν, ϰ, γ, the dimension of the space d, and data of the problem should interact each other for the existence of weak solutions to the problem. We also establish the conditions for uniqueness of the solutions to this problem.

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Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms

Cited by 17 publications

References 13 publications

ASYMTOTIC PROPERTIES OF THE GENERALIZED SOLUTION OF NONLINEAR PROBLEM FOR THE PSEUDOPARABOLIC EQUATION WITH p,q-LAPLACIAN STRUCTURE

ASYMTOTIC PROPERTIES OF THE GENERALIZED SOLUTION OF NONLINEAR PROBLEM FOR THE PSEUDOPARABOLIC EQUATION WITH p,q-LAPLACIAN STRUCTURE

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

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