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In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly 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 on the weight of the Sobolev spaces is considered.

Abstract. We suggest in this article a new explicit algorithm allowing to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to nonautonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting finite-dimensional time-dependent set in the phase space. In particular, this result shows that, for a wide class of nonautonomous equations of mathematical physics, the limit dynamics remains finite-dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this article on the model example of a nonautonomous reaction-diffusion system in a bounded domain.Introduction.

Our aim in this article is to give a construction of exponential attractors that are continuous under perturbations of the underlying semigroup. We note that the continuity is obtained without time shifts as it was the case in previous studies. Moreover, we obtain an explicit estimate for the symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter. As an application, we prove the continuity of exponential attractors for a viscous Cahn-Hilliard system to an exponential attractor for the limit Cahn-Hilliard system.

A nonlinear, density-dependent system of diffusion-reaction equations\ud describing development of bacterial biofilms is analyzed. It comprises two\ud non-standard diffusion effects, degeneracy as in the porous medium equation\ud and fast diffusion. The existence of a unique bounded solution and a global\ud attractor is proved in dependence of the boundary conditions. This is achieved\ud by studying a system of non-degenerate auxiliary approximation equations and\ud the construction of a Lipschitz continuous semigroup by passing to the limit\ud in the approximation parameter. Numerical examples are included in order to\ud illustrate the main result