Messoud Efendiev

Exponential attractors for a nonlinear reaction-diffusion system in R3

Abstract We give in this Note a construction of exponential attractors for a class of operators in Banach spaces (and not in Hilbert spaces only as it is the case for the classical constructions). We then apply this result to a reaction-diffusion system in R 3 .

Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Holder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous 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 paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

Lower Estimate of the Attractor Dimension for a Chemotaxis Growth System

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

Existence and longtime behavior of a biofilm model

Chapter 5 is devoted to biofilm modelling (meso-scale level), analysis and simulation which is one of the most active areas in modern microbiology. To this end it is enough to refer to: “It is the best of times for biofilm research” (Nature 76, vol. 15, pp. 76–81, 2007). In contrast to existing biofilm models, which are based mostly on discrete rules or hybrid models, we are mainly interested in a deterministic and continuous model which is described by PDEs. Chapter 5 consists of five sections. The first two Sects. 5.1 and 5.5 are concerned with the single species/single substrate models. In Sect. 5.1 we derive governing equations which describe spatial spreading mechanisms of biomass.The feature of these equations is that they are highly nonlinear density-dependent degenerate reaction-diffusion systems comprising two kind of degeneracy: porous medium and fast diffusion. We prove the well-posedness of the obtained equations and study the long-time dynamics of their solutions in terms of a global attractor. Moreover we analyze dependence of solutions on boundary conditions. Our numerical simulations of derived equations lead to mushroom patterns which were observed in the experimental studies.

On a new dimension estimate of the global attractor for chemotaxis-growth systems

In this paper we continue systematic study of the dimension e stimate of the global attractor for the chemotaxis-growth system. Using n onnegativity of solutions we manage significantly to improve dimension estimate s with respect to the chemotactic parameter.

The dimension of the global attractor for dissipative reaction-diffusion systems

Abstract Our aim in this note is to construct an exponential attractor of optimal (with respect to the dissipation parameter) fractal dimension for dissipative reaction-diffusion systems without conditions on the growth of the nonlinear term.

Nonlinear Riemann-Hilbert problems for doubly connected domains and closed boundary data

In this paper, for nonlinear Riemann–Hilbert problems in doubly connected domains with smooth as well as Lipschitz continuous boundary data, existence of at least two topologically different solutions is established. The main tools are the topological degree of quasi-ruled Fredholm mappings, Montel’s theorem, a priori estimates and the employment of classical modular function theory.

Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation

In this paper we study the long-time behaviour of solutions of reaction-diffusion equations (RDEs) in unbounded domains of Rn. In particular, we prove that, under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess infinite-dimensional exponential attractors whose Kolmogorov ε-entropy satisfies an estimate of the same type as that obtained previously for the ε-entropy of the global attractor. Moreover, we also study the problem of the approximation of these infinite-dimensional exponential attractors by finite-dimensional ones associated with the same RDEs in bounded domains.

Exponential attractor for an adsorbate-induced phase transition model in non smooth domain

We improve our preceding result obtained in Tsujikawa and Ya gi [10]. We construct the similar exponential attractors to the same adsor bate-induced phase transition model as in [10] but in a convex domain by using the compac t smoothing property of corresponding nonlinear semigroup. In [10], the dom ain has been assumed to haveC3 regularity to ensure the squeezing property of semigroup.

DIMENSION ESTIMATE OF THE EXPONENTIAL ATTRACTOR FOR THE CHEMOTAXIS–GROWTH SYSTEM

In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis–growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers. 2000 Mathematics Subject Classification. 35K15, 35K57, 37L30.