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1

Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators

Yangari M.

Journal of Applied Analysis

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2016

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Vol. 22, nr 1

55--65

EN

The aim of this paper is to study the large-time behaviour of mild solutions to the one-dimensional cooperative systems with anomalous diffusion when at least one entry of the initial condition decays slower than a power. We prove that the solution moves at least exponentially fast as time goes to infinity. Moreover, the exponent of propagation depends on the decay of the initial condition and of the reaction term.

2

Mechanical effects coupled with calcium waves

Kaźmierczak B., Dyzma M.

Archives of Mechanics

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2010

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Vol. 62, nr 2

121-133

EN

In the paper we find explicit formulae for heteroclinic travelling wave solutions in the system of equations describing the dynamics of cytosolic calcium concentration and the accompanying mechanical phenomena.

3

Diffusion of calcium in biological tissues and accompanying mechano-chemical effects

Peradzyński Z.

Archives of Mechanics

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2010

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Vol. 62, nr 6

423-440

EN

In this paper we consider the coupling between chemical and mechanical effects accompanying the diffusion of calcium, either in biological tissues or in a single long cell. The tissue is treated either as a 3-D, or as a quasi-2-D thin layer, of visco-elastic medium, whereas the cell is represented as a thin long cylinder. In particular, the influence of viscosity on the properties of calcium travelling waves is studied. In principle, we explore here the simplest model of calcium diffusion which is based on an effective diffusion coefficient, thus neglecting the details of the role played by buffers. The mechano-chemical coupling in the model is realized by the presence of a traction tensor, in addition to the viscoelastic stress tensor in the mechanical equations, and the strain tensor in the source term of the calcium diffusion equation, as proposed in [1–4]. Our aim is to provide a simple and effective theory, which can be useful in studying various effects influencing propagation of calcium waves. Since in the absence of viscosity the whole mechano-chemical system for calcium and buffers is easily reduced to the “chemical one”, i.e. it consists only of reaction diffusion equations, therefore we decided to perform expansion with respect to the viscosity. Treating, thus, viscous forces as a perturbation, we reduce the problem in each case to a single reaction diffusion equation for the calcium concentration. In this way we avoid the question of the existence of travelling wave solutions as for the so obtained models, their existence follows simply from already known theorems [5–9].

4

Three-dimensional Cellular Automata for Reaction-Diffusion Systems

Weimar J.R.

Fundamenta Informaticae

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2002

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Vol. 52, nr 1-3

277-284

EN

Cellular automata for reaction-diffusion systems are efficient enough to make the simulation of large three-dimensional systems feasible. The principal construction mechanisms used here are not much different from those for two-dimensional cellular automata. Diffusion is realized through a local averaging, and all the nonlinear reaction terms are collected in a table-lookup. The special issue in three dimensions is the need to increase time- and space- scales as much as possible to achieve sufficient system sizes. This can be done through the use of numerical integration schemes for constructing the lookup table, and through the use of special diffusion operators. We present examples of complex three-dimensional behaviors in an excitable reaction-diffusion system and in a model of a pattern-forming chemical reaction.

5

Global existence for a class of reaction-diffusion systems

Kouachi S., Youkana A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 3

303-308

EN

The purpose of this paper is to give sufficient conditions guaranteeing global existence, uniqueness and uniform boundedness of solutions for a class of reaction-diffusion systems.