Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity
Messoud Efendiev Alain Miranville
Discrete & Continuous Dynamical Systems - A 1999, 5(2): 399-424 doi: 10.3934/dcds.1999.5.399
Our aim in this article is to study the long time behavior of a class of reaction-diffusion equations in the whole space for which the nonlinearity depends explicitly on the gradient of the unknown function. We prove the existence of the global attractor and of exponential attractors for the semigroup associated with the equation. We also consider the nonautonomous case, and when the forcing term depends quasiperiodically on the time, we prove the existence of uniform and uniform exponential attractors.
keywords: unbounded domains exponential attractor Reaction-diffusion equations global attractor uniform attractor uniform exponential attractor. nonautonomous system
On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis
Messoud Efendiev Anna Zhigun
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 651-673 doi: 10.3934/dcds.2018028

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.

keywords: Attractor biofilm chemotaxis degenerate diffusion longtime dynamics
Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system
Messoud Efendiev Etsushi Nakaguchi Wolfgang L. Wendland
Conference Publications 2007, 2007(Special): 334-343 doi: 10.3934/proc.2007.2007.334
We study a finite-element approximation of the chemotaxis-growth system. We establish dimension estimate of global attractors for the approximate systems. Our results show that the estimates are uniform with respect to the discretization parameter and polynomial order with respect to the chemotactic coefficient in the equation.We especially emphasize that, this is just the same order (polynomial) as for the original system obtained in the preceding papers [Adv.Math.Sci.Appl. Part I and II].
keywords: finite-element approximation Dimension of global attractor chemotaxis-growth system.
Mathematical analysis of an in vivo model of mitochondrial swelling
Messoud Efendiev Mitsuharu Ôtani Hermann J. Eberl
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 4131-4158 doi: 10.3934/dcds.2017176

We analyze the effect of Robin boundary conditions in a mathematical model for a mitochondria swelling in a living organism. This is a coupled PDE/ODE model for the dependent variables calcium ion contration and three fractions of mitochondria that are distinguished by their state of swelling activity. The model assumes that the boundary is a permeable 'membrane', through which calcium ions can both enter or leave the cell. Under biologically relevant assumptions on the data, we prove the well-posedness of solutions of the model and study the asymptotic behavior of its solutions. We augment the analysis of the model with computer simulations that illustrate the theoretically obtained results.

keywords: PDE/ODE coupling longtim dynamics Robin boundary conditions mitochondria partial and complete swelling

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