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Abstract parabolic problems in ordered Banach spaces

Alexandre N. Carvalho, Jan W. Cholewa, Tomasz Dlotko (2001)

Colloquium Mathematicae

We consider abstract parabolic problems in ordered Banach spaces and give conditions under which they have global attractors. Our approach is via comparison of solutions. Within this approach abstract comparison principles are obtained and bounds on the attractors are given by order intervals in Banach spaces. These results are applied to ordinary differential equations and to parabolic equations for which the main part is given by a sum of fractional powers of sectorial operators having increasing...

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Alain Miranville (2014)

Open Mathematics

Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

Asymptotic dynamics in double-diffusive convection

Mikołaj Piniewski (2008)

Applicationes Mathematicae

We consider the double-diffusive convection phenomenon and analyze the governing equations. A system of partial differential equations describing the convective flow arising when a layer of fluid with a dissolved solute is heated from below is considered. The problem is placed in a functional analytic setting in order to prove a theorem on existence, uniqueness and continuous dependence on initial data of weak solutions in the class ( [ 0 , ) ; H ) L ² l o c ( + ; V ) . This theorem enables us to show that the infinite-dimensional...

Attractor of a semi-discrete Benjamin-Bona-Mahony equation on ℝ¹

Chaosheng Zhu (2015)

Annales Polonici Mathematici

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor τ in H¹(ℝ¹). In addition, we show that the global attractor τ is regular, i.e., τ is actually...

Attractors for general operators

Alain Miranville (2003)

Applications of Mathematics

In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.

Bifurcations of invariant measures in discrete-time parameter dependent cocycles

Anastasia Maltseva, Volker Reitmann (2015)

Mathematica Bohemica

We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent...

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