Mostafa Bendahmane

ANALYSIS OF A CLASS OF DEGENERATE REACTION-DIFFUSION SYSTEMS AND THE BIDOMAIN MODEL OF CARDIAC TISSUE

We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-diffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method.

Renormalized Entropy Solutions for Quasi-linear Anisotropic Degenerate Parabolic Equations

We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L data. This paper complements the work by Chen and Perthame [19], who developed a pure L theory based on the notion of kinetic solutions.

ANALYSIS OF A FINITE VOLUME METHOD FOR A CROSS-DIFFUSION MODEL IN POPULATION DYNAMICS

The main goal of this work is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time

ON A TWO-SIDEDLY DEGENERATE CHEMOTAXIS MODEL WITH VOLUME-FILLING EFFECT

L^1

Finite volume methods for degenerate chemotaxis model

compactness argument that mimics the compactness lemma due to S.N.~Kruzhkov. The proofs of these results are given in the Appendix.

Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis

We consider a fully parabolic model for chemotaxis with volume-filling effect and a nonlinear diffusion that degenerates in a two-sided fashion. We address the questions of existence of weak solutions and of their regularity by using, respectively, a regularization method and the technique of intrinsic scaling.

CONVERGENCE OF DISCRETE DUALITY FINITE VOLUME SCHEMES FOR THE CARDIAC BIDOMAIN MODEL

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection-diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.

On 3D DDFV Discretization of Gradient and Divergence Operators:Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

In this paper, we consider a system of nonlinear partial differential equations modeling the Lotka Volterra interactions of preys and actively moving predators with prey-taxis and spatial diffusion. The interaction between predators are modelized by the statement of a food pyramid condition. We establish the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. This paper is a generalization of the results obtained in [2].

ON A DOUBLY NONLINEAR DIFFUSION MODEL OF CHEMOTAXIS WITH PREVENTION OF OVERCROWDING

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model

Abstract. We present a detailed survey of discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W1,p compactness, discrete compactness in space and in time) for the so-called Discrete Duality Finite Volume (DDFV) schemes in three space dimensions. We concentrate mainly on the 3D CeVe-DDFV scheme presented in [IMA J. Numer. Anal., 32 (2012), pp. 1574–1603]. Some of our results are new, such as a general time-compactness result based upon the idea of Kruzhkov (1969); others generalize the ideas known for the 2D DDFV schemes or for traditional two-point-flux finite volume schemes. We illustrate the use of these tools by studying convergence of discretizations of nonlinear elliptic-parabolic problems of Leray–Lions kind, and provide numerical results for this example.