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1
Averaging of a nonlinear bipolar model with phase transition and oscillating external forces
Medjo T. T.
Journal of Applied Analysis
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2015
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Vol. 21, nr 2
123--146
EN
In this article, we consider a non-autonomous nonlinear bipolar with phase transition in a two-dimensional bounded domain. We assume that the external force is singularly oscillating and depends on a small parameter ε. We prove the existence of the uniform global attractor Aε. Furthermore, using the method of [9] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of Aε as e goes to zero.
2
Content available remote Global Attractor for a Class of Parabolic Equations with Infinite Delay
Anh C. T., Hieu L. V., Thanh D. T. P.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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Vol. 62, no 1
49--60
EN
We prove the existence of a compact connected global attractor for a class of abstract semilinear parabolic equations with infinite delay.
3
Content available remote Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on RN
Anh C. T., Thuy L. T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2013
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Vol. 61, no 1
47--65
EN
We prove the existence of global attractors for the following semilinear degenerate parabolic equation on RN: ∂u/∂t−div(σ(x)∇u)+λu+f(x,u)=g(x), under a new condition concerning the variable nonnegative diffusivity σ(⋅) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method.
4
Content available remote Global Attractor for a Fourth-Order Parabolic Equation Modeling Epitaxial Thin Film Growth
Duan N., Zhao X.
Bulletin of the Polish Academy of Sciences. Mathematics
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2012
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Vol. 60, no 3
259--268
EN
This paper is concerned with a fourth-order parabolic equation which models epitaxial growth of nanoscale thin films. Based on the regularity estimates for semigroups and the classical existence theorem of global attractors, we prove that the fourth order parabolic equation possesses a global attractor in a subspace of H2, which attracts all the bounded sets of H2 in the H2-norm.
5
Content available remote Global attractor for the convective Cahn-Hilliard equation in Hk
Zhao X., Duan N.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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Vol. 59, no 1
53-64
EN
We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in Hk.
6
Content available remote Global attractor for the convective Cahn-Hilliard equation
Zhao X., Liu C.
Bulletin of the Polish Academy of Sciences. Mathematics
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2010
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Vol. 58, no 2
117-127
EN
This paper is concerned with the convective Cahn-Hilliard equation. We use a classical theorem on existence of a global attractor to derive that the convective Cahn-Hilliard equation possesses a global attractor on some subset of H2.
7
Content available remote Dynamically equivalent perturbations of linear parabolic equations
Czaja R.
Demonstratio Mathematica
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2004
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Vol. 37, nr 2
327--348
EN
A family of abstract parabolic equations with sectorial operator is studied in this paper. The conditions are provided to show that the global attractors for each equation exist and coincide. Although the common dynamics is simple, the examples presented in the final part of the paper indicate that the considered family may contain a linear equation together with a large number of its nonlinear perturbations. The mentioned examples include both scalar second order equations and the celebrated Cahn-Hilliard system.
8
Content available remote Hyperbolic equations in uniform spaces
Cholewa J. W., Dlotko T.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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Vol. 52, nr 3
249-263
EN
The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of R^n. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space H[...]^R^n) x L[...](R^n). Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.
9
Content available remote Asymptotics of pseudodifferential parabolic equations
Cholewa J. W., Dłotko T., Turski A. W.
Demonstratio Mathematica
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2002
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Vol. 35, nr 1
74-91
EN
The paper provides new type examples covered by the general theory of global attractors for abstract parabolic equations presented in the monograph [C-D 1]. Inside the class of sectorial equations of the form (1) u+Au = F(u), t > 0, u(0) = uo, we cover pseudodifferential parabolic problems (2) m = -(-A)u + f(u), a należy (0,1), studied with suitable initial-boundary conditions and also their generalizations to problems with the main part being a finite sum of the fractional powers.
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