Sergei Shmarev - Academia.edu (original) (raw)

Papers by Sergei Shmarev

Research paper thumbnail of Chapter II: Equivalence transformations of evolution equations

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Chapter I: The Verigin problem

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Instantaneous appearance of singularities of a solution of a degenerate parabolic equation

Siberian Mathematical Journal, 1991

Research paper thumbnail of On the Boundary Layer for Dilatant Fluids

Energy Methods in Continuum Mechanics, 1996

ABSTRACT

Research paper thumbnail of A compactness lemma of Aubin type and its application to degenerate parabolic equations

Let Ω ⊂ R n be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M ⊂ L 2 (0, T ; W 1 ... more Let Ω ⊂ R n be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M ⊂ L 2 (0, T ; W 1 2 (Ω)) ∩ L∞(Ω × (0, T)) is bounded and the set {∂tΦ(v)| v ∈ M} is bounded in L 2 (0, T ; W −1 2 (Ω)), then there is a sequence {v k } ∈ M such that v k v ∈ L 2 (0, T ; W 1 2 (Ω)), and v k → v, Φ(v k) → Φ(v) a.e. in Ω T = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solution.

Research paper thumbnail of Chapter IV: Parabolic equations in several space dimensions

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Evolution Equations and Lagrangian Coordinates

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Chapter III: One-dimensional parabolic equations

Research paper thumbnail of Extinction of solutions of parabolic equations with variable anisotropic nonlinearities

Proceedings of the Steklov Institute of Mathematics, 2008

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anis... more We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional p(x, t)-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as t → ∞.

Research paper thumbnail of On the blow-up of solutions to anisotropic parabolic equations with variable nonlinearity

Proceedings of the Steklov Institute of Mathematics, 2010

ABSTRACT The aim of this paper is to establish sufficient conditions of the finite time blow-up i... more ABSTRACT The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } . Two different cases are studied. In the first case a i ≡ a i (x), p i ≡ 2, σ i ≡ σ i (x, t), and b i (x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σ j (x, t) > 2 and either b j > 0, or b j (x, t) ≥ 0 and Σπ b j −ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ j . In the case of the quasilinear equation with the exponents p i and σ i depending only on x, we show that the solutions may blow up if min σ i ≥ max p i , b i ≥ 0, and there exists at least one j for which min σ j > max p j and b j > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (b i ≤ 0) and reaction terms.

Research paper thumbnail of Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

Journal of Mathematical Analysis and Applications, 2010

We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem ... more We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equationsut−∑i=1nDi(ai(x,t,u)|Diu|pi(x,t)−2Diu)+c(x,t)|u|σ(x,t)−2u=f(x,t) with variable exponents of nonlinearity pi(x,t),σ(x,t)∈(1,∞). We show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of

Research paper thumbnail of Lagrangian approach to the study of level sets II: A quasilinear equation in climatology

Journal of Mathematical Analysis and Applications, 2009

ABSTRACT We study the dynamics and regularity of the level sets in solutions of the semilinear pa... more ABSTRACT We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation where Ω⊂Rn is a ring-shaped domain, Δpu is the p-Laplace operator, a and μ are given positive constants, and H(⋅) is the Heaviside maximal monotone graph: H(s)=1 if s>0, H(0)=[0,1], H(s)=0 if s<0. The mathematical models of this type arise in climatology, the case p=3 was proposed and justified by P. Stone in 1972. We establish the conditions on the initial data which guarantee that the level sets are hypersurfaces, study the regularity of Γμ(t) and derive the differential equation that governs the dynamics of Γμ(t). The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Γμ(t) into a stationary one.

Research paper thumbnail of Doubly degenerate parabolic equations with variable nonlinearity II: Blow-up and extinction in a finite time

Nonlinear Analysis: Theory, Methods & Applications, 2014

Research paper thumbnail of Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions

Communications on Pure and Applied Analysis, 2012

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonsta... more The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div (a(x, t, u)|u|(x, t)|u|p(x, t)-2 with given variable exponents (x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.

Research paper thumbnail of Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics

Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz ... more Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz S. Shmarev Energy Methods for Free Boundary Problems Applications to Nonlinear PDEs and Fluid Mechanics Birkhauser Page 2. Page 3. s On EU1U-ST6-E3R1 Page 4. ...

Research paper thumbnail of Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics

Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz ... more Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz S. Shmarev Energy Methods for Free Boundary Problems Applications to Nonlinear PDEs and Fluid Mechanics Birkhauser Page 2. Page 3. s On EU1U-ST6-E3R1 Page 4. ...

Research paper thumbnail of Existence and uniqueness for doubly nonlinear parabolic equations with nonstandard growth conditions

Differential Equations & Applications, 2012

Research paper thumbnail of Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension

Trends in Partial Differential Equations of Mathematical Physics, 2005

Research paper thumbnail of Lagrangian coordinates in free boundary problems for multidimensional parabolic equations

Elliptic and Parabolic Problems, 2002

Research paper thumbnail of Interfaces in solutions of diffusion-absorption equations

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a Matematicas, 2002

We study the properties of interfaces in solutions of the Cauchy problem for the nonlinear degene... more We study the properties of interfaces in solutions of the Cauchy problem for the nonlinear degenerate parabolic equation ut = ∆ u m − u p in R n × (0, T ] with the parameters m > 1, p > 0 satisfying the condition m + p ≥ 2. We show that the velocity of the interface

Research paper thumbnail of Chapter II: Equivalence transformations of evolution equations

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Chapter I: The Verigin problem

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Instantaneous appearance of singularities of a solution of a degenerate parabolic equation

Siberian Mathematical Journal, 1991

Research paper thumbnail of On the Boundary Layer for Dilatant Fluids

Energy Methods in Continuum Mechanics, 1996

ABSTRACT

Research paper thumbnail of A compactness lemma of Aubin type and its application to degenerate parabolic equations

Let Ω ⊂ R n be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M ⊂ L 2 (0, T ; W 1 ... more Let Ω ⊂ R n be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M ⊂ L 2 (0, T ; W 1 2 (Ω)) ∩ L∞(Ω × (0, T)) is bounded and the set {∂tΦ(v)| v ∈ M} is bounded in L 2 (0, T ; W −1 2 (Ω)), then there is a sequence {v k } ∈ M such that v k v ∈ L 2 (0, T ; W 1 2 (Ω)), and v k → v, Φ(v k) → Φ(v) a.e. in Ω T = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solution.

Research paper thumbnail of Chapter IV: Parabolic equations in several space dimensions

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Evolution Equations and Lagrangian Coordinates

Evolution Equations and Lagrangian Coordinates, 1997

Research paper thumbnail of Chapter III: One-dimensional parabolic equations

Research paper thumbnail of Extinction of solutions of parabolic equations with variable anisotropic nonlinearities

Proceedings of the Steklov Institute of Mathematics, 2008

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anis... more We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional p(x, t)-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as t → ∞.

Research paper thumbnail of On the blow-up of solutions to anisotropic parabolic equations with variable nonlinearity

Proceedings of the Steklov Institute of Mathematics, 2010

ABSTRACT The aim of this paper is to establish sufficient conditions of the finite time blow-up i... more ABSTRACT The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } . Two different cases are studied. In the first case a i ≡ a i (x), p i ≡ 2, σ i ≡ σ i (x, t), and b i (x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σ j (x, t) > 2 and either b j > 0, or b j (x, t) ≥ 0 and Σπ b j −ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ j . In the case of the quasilinear equation with the exponents p i and σ i depending only on x, we show that the solutions may blow up if min σ i ≥ max p i , b i ≥ 0, and there exists at least one j for which min σ j > max p j and b j > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (b i ≤ 0) and reaction terms.

Research paper thumbnail of Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

Journal of Mathematical Analysis and Applications, 2010

We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem ... more We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equationsut−∑i=1nDi(ai(x,t,u)|Diu|pi(x,t)−2Diu)+c(x,t)|u|σ(x,t)−2u=f(x,t) with variable exponents of nonlinearity pi(x,t),σ(x,t)∈(1,∞). We show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of

Research paper thumbnail of Lagrangian approach to the study of level sets II: A quasilinear equation in climatology

Journal of Mathematical Analysis and Applications, 2009

ABSTRACT We study the dynamics and regularity of the level sets in solutions of the semilinear pa... more ABSTRACT We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation where Ω⊂Rn is a ring-shaped domain, Δpu is the p-Laplace operator, a and μ are given positive constants, and H(⋅) is the Heaviside maximal monotone graph: H(s)=1 if s>0, H(0)=[0,1], H(s)=0 if s<0. The mathematical models of this type arise in climatology, the case p=3 was proposed and justified by P. Stone in 1972. We establish the conditions on the initial data which guarantee that the level sets are hypersurfaces, study the regularity of Γμ(t) and derive the differential equation that governs the dynamics of Γμ(t). The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Γμ(t) into a stationary one.

Research paper thumbnail of Doubly degenerate parabolic equations with variable nonlinearity II: Blow-up and extinction in a finite time

Nonlinear Analysis: Theory, Methods & Applications, 2014

Research paper thumbnail of Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions

Communications on Pure and Applied Analysis, 2012

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonsta... more The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div (a(x, t, u)|u|(x, t)|u|p(x, t)-2 with given variable exponents (x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.

Research paper thumbnail of Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics

Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz ... more Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz S. Shmarev Energy Methods for Free Boundary Problems Applications to Nonlinear PDEs and Fluid Mechanics Birkhauser Page 2. Page 3. s On EU1U-ST6-E3R1 Page 4. ...

Research paper thumbnail of Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics

Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz ... more Page 1. Progress in Nonlinear Differential Equations and Their Applications SN Antontsev JI Dlaz S. Shmarev Energy Methods for Free Boundary Problems Applications to Nonlinear PDEs and Fluid Mechanics Birkhauser Page 2. Page 3. s On EU1U-ST6-E3R1 Page 4. ...

Research paper thumbnail of Existence and uniqueness for doubly nonlinear parabolic equations with nonstandard growth conditions

Differential Equations & Applications, 2012

Research paper thumbnail of Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension

Trends in Partial Differential Equations of Mathematical Physics, 2005

Research paper thumbnail of Lagrangian coordinates in free boundary problems for multidimensional parabolic equations

Elliptic and Parabolic Problems, 2002

Research paper thumbnail of Interfaces in solutions of diffusion-absorption equations

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie a Matematicas, 2002

We study the properties of interfaces in solutions of the Cauchy problem for the nonlinear degene... more We study the properties of interfaces in solutions of the Cauchy problem for the nonlinear degenerate parabolic equation ut = ∆ u m − u p in R n × (0, T ] with the parameters m > 1, p > 0 satisfying the condition m + p ≥ 2. We show that the velocity of the interface