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Global Existence and Boundedness of Solutions to a Model of Chemotaxis

J. Dyson, R. Villella-Bressan, G. F. Webb (2008)

Mathematical Modelling of Natural Phenomena

A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are L∞-bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

Global existence and convergence to steady states in a chemorepulsion system

Tomasz Cieślak, Philippe Laurençot, Cristian Morales-Rodrigo (2008)

Banach Center Publications

In this paper we consider a model of chemorepulsion. We prove global existence and uniqueness of smooth classical solutions in space dimension n = 2. For n = 3,4 we prove the global existence of weak solutions. The convergence to steady states is shown in all cases.

Global existence of solutions to a chemotaxis system with volume filling effect

Tomasz Cieślak (2008)

Colloquium Mathematicae

A system of quasilinear parabolic equations modelling chemotaxis and taking into account the volume filling effect is studied under no-flux boundary conditions. The resulting system is non-uniformly parabolic. A Lyapunov functional for the system is constructed. The proof of existence and uniqueness of regular global-in-time solutions is given in cases when either the Lyapunov functional is bounded from below or chemotactic forces are suitably weakened. In the first case solutions are uniformly...

Group actions on monotone skew-product semiflows with applications

Feng Cao, Mats Gyllenberg, Yi Wang (2016)

Journal of the European Mathematical Society

We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group G -action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of...

Homogenization of a carcinogenesis model with different scalings with the homogenization parameter

Isabell Graf, Malte A. Peter (2014)

Mathematica Bohemica

In the context of periodic homogenization based on two-scale convergence, we homogenize a linear system of four coupled reaction-diffusion equations, two of which are defined on a manifold. The system describes the most important subprocesses modeling the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum, which constitutes a fine structure inside the cell....

Homogenization of periodic non self-adjoint problems with large drift and potential

Grégoire Allaire, Rafael Orive (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by ε the period, the potential or zero-order term is scaled as ε - 2 and the drift or first-order term is scaled as ε - 1 . Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the...

Implementation of the MR tractography visualization kit based on the anisotropic Allen-Cahn equation

Pavel Strachota (2009)

Kybernetika

Magnetic Resonance Diffusion Tensor Imaging (MR–DTI) is a noninvasive in vivo method capable of examining the structure of human brain, providing information about the position and orientation of the neural tracts. After a short introduction to the principles of MR–DTI, this paper describes the steps of the proposed neural tract visualization technique based on the DTI data. The cornerstone of the algorithm is a texture diffusion procedure modeled mathematically by the problem for the Allen–Cahn...

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