S. Antontsev - Academia.edu (original) (raw)

Papers by S. Antontsev

We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Lapla... more We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Laplacian, ut −∆p(x)u− Z t 0 g(t− s)∆p(x)u(x, s) ds ∈ F(u) in QT = Ω× (0, T ), where Ω ⊂ Rn, n ≥ 1, is a bounded domain with Lipschitz-continuous boundary. The exponent p(x) is a given measurable function, p− ≤ p(x) ≤ p+ a.e. in Ω for some bounded constants p− > max{1, 2n n+2 } and p+ <∞. It is assumed that g, g′ ∈ L2(0, T ), and that the multivalued function F(·) is globally Lipschitz, has convex closed values and F(0) 6= ∅. We prove that the homogeneous Dirichlet problem has a local in time weak solution. Also we show that when p− > 2 and uF(u) ⊆ {v ∈ L2(Ω) : v ≤ u2 a.e. in Ω} with a sufficiently small > 0 the weak solution possesses the property of finite speed of propagation of disturbances from the initial data and may exhibit the waiting time property. Estimates on the evolution of the null-set of the solution are presented.

Siberian Mathematical Journal, 1997

Complex Variables and Elliptic Equations, 2011

This special issue is dedicated to Professor V.V. Zhikov, a brilliant Russian mathematician, on t... more This special issue is dedicated to Professor V.V. Zhikov, a brilliant Russian mathematician, on the occasion of his 70th anniversary. Research topics of Professor Zhikov run from almost-periodic functions to homogenization, to spectral theory, to calculus of variations – to mention a few. In particular, his pioneering works and deep results on variational problems with non-standard Lagrangians defined on Sobolev–Orlicz spaces largely determined the development of the field chosen as the topic of this special volume. A full review of V.V. Zhikov’s scientific achievements is beyond the scope of this introduction. Let us only mention that his early works were dedicated to the study of almost-periodic functions and operator equations. Thereafter, V.V. Zhikov’s research interests shifted to the homogenization theory and non-linear analysis. His study of homogenization problems brought him to consider elliptic equations with non-standard conditions of coerciveness and growth. It is generally accepted that V.V. Zhikov was the first who considered elliptic equations with a variable nonlinearity exponent in connection with homogenization problems for nonlinear

SIAM Journal on Numerical Analysis, 2013

In this work, we study the convergence of the finite element method when applied to the following... more In this work, we study the convergence of the finite element method when applied to the following parabolic equation: ut = div(|u| γ(x) ∇u) + f (x, t), x ∈ Ω ⊂ R m , t ∈]0, T ]. Since the problem may be of degenerate type, we utilize an approximate problem, regularized by introducing a parameter ε. We prove, under certain conditions on γ and f , that the weak solution of the approximate problem converges to the weak solution of the initial problem, when the parameter ε tends to zero. Discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved. Finally, we present some numerical results of a MATLAB implementation of the method.

Computers & Mathematics with Applications, 2017

Journal of Mathematical Fluid Mechanics, 2009

Journal of Applied Mechanics and Technical Physics, 2008

We present an overview of the recent advances in the theory of parabolic equations with nonstanda... more We present an overview of the recent advances in the theory of parabolic equations with nonstandard anisotropic growth conditions. The presentation includes the existence theorems in the variable exponents Sobolev spaces and a description of the properties of propagation of disturbances from the data, intrinsic for solutions of such equations.

Nonlinear Analysis: Theory, Methods & Applications, 2009

Differential and Integral Equations, 2014

Asymptotic Analysis

A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic di... more A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473(2) (2019) 1122–1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.

Nonlinear Analysis: Real World Applications

Quarterly of Applied Mathematics

Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizab... more Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizable by a direct electric current for use in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of cross-linked polyelectrolyte, weakly charged with the same sign as the membrane’s charge. The present paper is concerned with this effect. In order to explain this feature, a simple limiting ion-exchange ’funnel’ model of a modified membrane is derived from the original two-layer model. In this model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface, which itself has the same order as the bulk equation. Rigorous analysis of the ’funnel’ model shows that the value of the limiting current through a modified membrane, which is the m...

Applicable Analysis, 2015

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2009

We prove several uniform L 1-estimates on solutions of a general class of one-dimensional parabol... more We prove several uniform L 1-estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don't depend on the coefficients, nor on the size of the spatial domain. The estimates concern the own solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the L 1-norm of solutions, as t → +∞, is also given. Estimaciones sobre el gradiente y otras propiedades cualitativas de las soluciones de sistemas parabólicos no lineales no autónomos. Resumen. En este artículo se obtienen varias estimaciones uniformes en L 1 para las soluciones y su derivada espacial de ciertos sistemas parabólicos no lineales que pueden estar acoplados en los términos de difusión y que, de hecho, puede ser de tipo degenerado. Tales estimaciones son uniformes en el sentido de que no dependen de los coeficientes del sistema, ni del tamaño del dominio espacial. Las estimaciones se refieren a la norma L 1 de la propia solución o/y de su gradiente espacial. Este trabajo extiende, al caso de coeficientes no autónomos y a posibles condiciones de contorno no homogéneas, ciertos resultados previos de los autores. Además, se ofrece una aplicación al estudio del decaimiento de la norma L 1 de la solución, cuando t → +∞.

International Series of Numerical Mathematics, 2007

This book gathers a collection of refereed articles containing original results reporting the rec... more This book gathers a collection of refereed articles containing original results reporting the recent original contributions of the lectures and communications presented at the Free Boundary Problems (FBP2005) Conference that took place at the University of Coimbra, Portugal, from 7 to 12 of June 2005. They deal with the Mathematics of a broad class of models and problems involving nonlinear partial differential equations arising in Physics, Engineering, Biology and Finance. Among the main topics, the talks considered free boundary problems in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in image processing, in financial mathematics or in computations for inter-scale problems. FBP2005 was the 10th Conference of a Series started in 1981 in Montecatini, Italy, that has had a continuous development in the following conferences

We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Lapla... more We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Laplacian, ut −∆p(x)u− Z t 0 g(t− s)∆p(x)u(x, s) ds ∈ F(u) in QT = Ω× (0, T ), where Ω ⊂ Rn, n ≥ 1, is a bounded domain with Lipschitz-continuous boundary. The exponent p(x) is a given measurable function, p− ≤ p(x) ≤ p+ a.e. in Ω for some bounded constants p− > max{1, 2n n+2 } and p+ <∞. It is assumed that g, g′ ∈ L2(0, T ), and that the multivalued function F(·) is globally Lipschitz, has convex closed values and F(0) 6= ∅. We prove that the homogeneous Dirichlet problem has a local in time weak solution. Also we show that when p− > 2 and uF(u) ⊆ {v ∈ L2(Ω) : v ≤ u2 a.e. in Ω} with a sufficiently small > 0 the weak solution possesses the property of finite speed of propagation of disturbances from the initial data and may exhibit the waiting time property. Estimates on the evolution of the null-set of the solution are presented.

Siberian Mathematical Journal, 1997

Complex Variables and Elliptic Equations, 2011

This special issue is dedicated to Professor V.V. Zhikov, a brilliant Russian mathematician, on t... more This special issue is dedicated to Professor V.V. Zhikov, a brilliant Russian mathematician, on the occasion of his 70th anniversary. Research topics of Professor Zhikov run from almost-periodic functions to homogenization, to spectral theory, to calculus of variations – to mention a few. In particular, his pioneering works and deep results on variational problems with non-standard Lagrangians defined on Sobolev–Orlicz spaces largely determined the development of the field chosen as the topic of this special volume. A full review of V.V. Zhikov’s scientific achievements is beyond the scope of this introduction. Let us only mention that his early works were dedicated to the study of almost-periodic functions and operator equations. Thereafter, V.V. Zhikov’s research interests shifted to the homogenization theory and non-linear analysis. His study of homogenization problems brought him to consider elliptic equations with non-standard conditions of coerciveness and growth. It is generally accepted that V.V. Zhikov was the first who considered elliptic equations with a variable nonlinearity exponent in connection with homogenization problems for nonlinear

SIAM Journal on Numerical Analysis, 2013

In this work, we study the convergence of the finite element method when applied to the following... more In this work, we study the convergence of the finite element method when applied to the following parabolic equation: ut = div(|u| γ(x) ∇u) + f (x, t), x ∈ Ω ⊂ R m , t ∈]0, T ]. Since the problem may be of degenerate type, we utilize an approximate problem, regularized by introducing a parameter ε. We prove, under certain conditions on γ and f , that the weak solution of the approximate problem converges to the weak solution of the initial problem, when the parameter ε tends to zero. Discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved. Finally, we present some numerical results of a MATLAB implementation of the method.

Computers & Mathematics with Applications, 2017

Journal of Mathematical Fluid Mechanics, 2009

Journal of Applied Mechanics and Technical Physics, 2008

We present an overview of the recent advances in the theory of parabolic equations with nonstanda... more We present an overview of the recent advances in the theory of parabolic equations with nonstandard anisotropic growth conditions. The presentation includes the existence theorems in the variable exponents Sobolev spaces and a description of the properties of propagation of disturbances from the data, intrinsic for solutions of such equations.

Nonlinear Analysis: Theory, Methods & Applications, 2009

Differential and Integral Equations, 2014

Asymptotic Analysis

A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic di... more A nonlinear initial and boundary-value problem for the Kelvin–Voigt equations with anisotropic diffusion, relaxation and absorption/damping terms is considered in this work. The global and local unique solvability of the problem was established in (J. Math. Anal. Appl. 473(2) (2019) 1122–1154). In the present work, we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients should interact with the problem data for the solutions of this problem display exponential and polynomial time-decays. We also establish the conditions for the solutions of this problem to blow-up in a finite time in three different cases: problem without convection, full anisotropic problem, and the problem with isotropic relaxation.

Nonlinear Analysis: Real World Applications

Quarterly of Applied Mathematics

Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizab... more Inexpensive highly permselective heterogeneous ion exchange membranes are prohibitively polarizable by a direct electric current for use in electrodialysis. According to recent experiments, polarizability of these membranes may be considerably reduced by casting on their surface a thin layer of cross-linked polyelectrolyte, weakly charged with the same sign as the membrane’s charge. The present paper is concerned with this effect. In order to explain this feature, a simple limiting ion-exchange ’funnel’ model of a modified membrane is derived from the original two-layer model. In this model, asymptotically valid for a thin coating, solution of the ionic transport equations in it is replaced, via a suitable averaging procedure, by a single nonlinear boundary condition for the membrane/solution interface, which itself has the same order as the bulk equation. Rigorous analysis of the ’funnel’ model shows that the value of the limiting current through a modified membrane, which is the m...

Applicable Analysis, 2015

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2009

We prove several uniform L 1-estimates on solutions of a general class of one-dimensional parabol... more We prove several uniform L 1-estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don't depend on the coefficients, nor on the size of the spatial domain. The estimates concern the own solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the L 1-norm of solutions, as t → +∞, is also given. Estimaciones sobre el gradiente y otras propiedades cualitativas de las soluciones de sistemas parabólicos no lineales no autónomos. Resumen. En este artículo se obtienen varias estimaciones uniformes en L 1 para las soluciones y su derivada espacial de ciertos sistemas parabólicos no lineales que pueden estar acoplados en los términos de difusión y que, de hecho, puede ser de tipo degenerado. Tales estimaciones son uniformes en el sentido de que no dependen de los coeficientes del sistema, ni del tamaño del dominio espacial. Las estimaciones se refieren a la norma L 1 de la propia solución o/y de su gradiente espacial. Este trabajo extiende, al caso de coeficientes no autónomos y a posibles condiciones de contorno no homogéneas, ciertos resultados previos de los autores. Además, se ofrece una aplicación al estudio del decaimiento de la norma L 1 de la solución, cuando t → +∞.

International Series of Numerical Mathematics, 2007

This book gathers a collection of refereed articles containing original results reporting the rec... more This book gathers a collection of refereed articles containing original results reporting the recent original contributions of the lectures and communications presented at the Free Boundary Problems (FBP2005) Conference that took place at the University of Coimbra, Portugal, from 7 to 12 of June 2005. They deal with the Mathematics of a broad class of models and problems involving nonlinear partial differential equations arising in Physics, Engineering, Biology and Finance. Among the main topics, the talks considered free boundary problems in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in image processing, in financial mathematics or in computations for inter-scale problems. FBP2005 was the 10th Conference of a Series started in 1981 in Montecatini, Italy, that has had a continuous development in the following conferences