The attractor for a nonlinear reaction-diffusion system in an unbounded domain

In this paper the quasilinear second order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim in this article is to study the long-time behaviour of parabolic systems for which the nonlinearity depends explicitely on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors from weight of the Sobolev spaces are considered

Generalized necessary scaling condition and stability of chemical reactors with several educts

We present, for a class of industrially relevant chemical reactions with two educts the dependence of stability on important chemical parameters, such as coolant, dilution and diffusion rates. The main analytical tools are generalized upscaling balance condition for the equilibria concentrations and spectral properties of corresponding operators. Although we illustrate the stability analysis for a model reactor (2 educts, E1 and E2), it should be emphasized that our approach is applicable to more complex reaction mechanisms

The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity

This article is devoted to the study of the long-time behavior of solutions of the following 4th order parabolic system in a bounded smooth domain Ω ⊂⊂ ℝn: b∂t u = -δx (aδxu - a∂tu - f(u) + g̃), (1) where u = (u1,., uk) is an unknown vector-valued function, a and b are given constant matrices such that a + a* > 0, b = b* > 0, α > 0 is a positive number, and f and g are given functions. Note that the nonlinearity f is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for system (1) is proved under some natural assumptions on the nonlinear term f

The long-time behaviour of the thermoconvective flow in a porous medium

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations posesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system

On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function