Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity
Messoud Efendiev Alain Miranville
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

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
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

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
On an ODE-PDE coupling model of the mitochondrial swelling process
Sabine Eisenhofer Messoud A. Efendiev Mitsuharu Ôtani Sabine Schulz Hans Zischka
Mitochondrial swelling has huge impact to multicellular organisms since it triggers apoptosis, the programmed cell death. In this paper we present a new mathematical model of this phenomenon. As a novelty it includes spatial effects, which are of great importance for the in vivo process. Our model considers three mitochondrial subpopulations varying in the degree of swelling. The evolution of these groups is dependent on the present calcium concentration and is described by a system of ODEs, whereas the calcium propagation is modeled by a reaction-diffusion equation taking into account spatial effects. We analyze the derived model with respect to existence and long-time behavior of solutions and obtain a complete mathematical classification of the swelling process.
keywords: partial and complete swelling. Mitochondria long-time dynamics ODE-PDE coupling
A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions
Hassan Khassehkhan Messoud A. Efendiev Hermann J. Eberl
We study a mathematical model that describes how a "good" bacterial biofilm controls the growth of a harmful pathogenic bacterial biofilm. The underlying mechanism is a modification of the local protonated acid concentration, which in turn decreases the local pH and, thus, makes growth conditions for the pathogens less favorable, while the control-agent itself is more tolerant to these changes. This system is described by a system of 5 density-dependent diffusion-reaction equations that show two nonlinear diffusion effects: porous medium degeneracy and fast diffusion. This is a multi-species expansion of a previously studied single species biofilm model. In this paper we prove the existence of solutions to this model and show in numerical simulations the effectiveness of the control mechanism.
keywords: degeneracy simulation. nonlinear diffusion biofilm probiotic
Existence and longtime behavior of a biofilm model
Messoud A. Efendiev Sergey Zelik Hermann J. Eberl
A nonlinear, density-dependent system of diffusion-reaction equations describing development of bacterial biofilms is analyzed. It comprises two non-standard diffusion effects, degeneracy as in the porous medium equation and fast diffusion. The existence of a unique bounded solution and a global attractor is proved in dependence of the boundary conditions. This is achieved by studying a system of non-degenerate auxiliary approximation equations and the construction of a Lipschitz continuous semigroup by passing to the limit in the approximation parameter. Numerical examples are included in order to illustrate the main result.
keywords: biofilm global attractor Nonlinear diffusion degeneracy
Analysis of a degenerate biofilm model with a nutrient taxis term
Hermann J. Eberl Messoud A. Efendiev Dariusz Wrzosek Anna Zhigun
We introduce and analyze a prototype model for chemotactic effects in biofilm formation. The model is a system of quasilinear parabolic equations into which two thresholds are built in. One occurs at zero cell density level, the second one is related to the maximal density which the cells cannot exceed. Accordingly, both diffusion and taxis terms have degenerate or singular parts. This model extends a previously introduced degenerate biofilm model by combining it with a chemotaxis equation. We give conditions for existence and uniqueness of weak solutions and illustrate the model behavior in numerical simulations.
keywords: singular quasilinear parabolic equation compactness method. biofilm Chemotaxis fast diffusion degenerate diffusion

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