ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS

This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.

Dynamics of a Reaction-Diffusion System with Quiescence

In this paper,the dynamical behavior of a reaction-diffusion system with quiescence in a closed environment is investigated.The global existence of the solution is obtained by the upper and lower solution method,and the dissipative structure of the system is derived by constructing Lyapunov functions.

High accuracy non-equidistant method for singular perturbation reaction-diffusion problem

Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.

STOCHASTIC CRACKING AND HEALING BEHAVIORS OF THIN FILMS DURING REACTION-DIFFUSION GROWTH

The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.

GLOBAL STABILITY OF TRAVELING WAVEFRONTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts （waves with speeds c 〉 c＊, where c=c＊ is the minimal speed） is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].

Numerical Simulation of Reaction-Diffusion during Carburization of HK40 Steel

Two types of carbides M23C6 and M7C3 precipitate orderly as carbon concentration in a high Cr-Ni austenitic steel increases during carburization process. The mathematical model that describes diffusion of carbon and the precipitation of M23C6 and M7C3 has been studied. A criterion to judge when the transformation of M23C6 to M7C3 is over and M7C3 precipitates directly has been given in simulated calculation. By applying the model, the carburization of HK40 steel has been calculated by means of finite difference computation techniques. The pack carburization tests for the HK40 steel have been carried out at 1273 K. The comparison between the experimental and the calculated results show acceptable agreement.

Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

Finite volume element method for analysis of unsteady reaction-diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.

UNIFORM QUASI-DIFFERENTIABILITY OF SEMIGROUP TO NONLINEAR REACTION-DIFFUSION EQUATIONS WITH SUPERCRITI C AL EXPONENT

A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.

A Reaction-Diffusion Algorithm for Segmentation of Three-Dimensional Sinusoidal Networks in Rats Fed a High-Fat and High-Cholesterol Diet: New Insights and Evaluation

Microstructures in the liver are primarily composed of hepatocytes, hepatic blood, and biliary vessels. Because each hepatocyte comes in contact with both vessels, these vessels form three-dimensional (3D) periodic network patterns. Confocal microscope images are useful for observing 3D structures;however, it is necessary to explicitly describe the vessel structures using 3D images of sinusoidal endothelial cells. For this purpose, we propose a new approach for image segmentation based on the Turing reaction-diffusion model, in which temporal and spatial patterns are self-organized. Turing conditions provided reliable tools for describing the 3D structures. Moreover, using the proposed method, the sinusoidal patterns of rats fed a high-fat/high-cholesterol diet were examined;these rats exhibited pathological features similar to those of human patients with nonalcoholic steatohepatitis related to metabolic syndrome. The findings showed that the parameter in diffusion terms differed significantly among the experimental groups. This observation provided a heuristic argument for parameter selection leading to pattern recognition problems in diseased rats.

非自治Reaction-Diffusion方程的后向紧吸引子的正则性

运用后向Gronwall型不等式及一个后向截断的方法,在g是后向绝对连续的假设条件下,证明非自治Reaction-Diffusion方程在正则p(p≥2)次可积空间上存在唯一的后向紧拉回吸引子,这种紧性体现了非自治系统关于时间依赖的独特性,展示了非自治系统和自治系统的本质区别.

Simulating Some Complex Phenomena in Hydrothermal Ore-Forming Processes by Reaction-Diffusion CNN

Complexity phenomena like dynamic and static patterns, order from disorder, chaos and catastrophe were simulated by the application of 2-D reaction-diffusion CNN of two state variables and two diffusion coefficients transformed from Zhabotinksii model. They revealed somehow the mechanism of hydrothermal ore-forming processes, and answered several questions about the onset of ore forming.

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.

Reaction-Diffusion Algorithm for Quantitative Analysis of Periodic V-Shaped Bundles of Hair Cells in the Inner Ear

Columnar outer hair cells of the inner ear form V-shaped bundles, which are arranged in three rows in the cochlea. If the formation of these V-shaped bundles is disturbed, sensorineural hearing loss occurs. To quantify the distribution of V-shaped bundles, the use of a Turing-type reaction-diffusion (RD) system, including anisotropic diffusion, should be considered. We found that a periodic triangle pattern appears based on the anisotropy in the RD system. Then, using the proposed RD system, the image of the V-shaped bundle was examined. The results showed that the correlation between the RD pattern and V-shape bundles not only classifies the experimental result but also significantly differs in the number of directional changes in a set of V-shape bundle. Therefore, this process can help quantify the images of V-shaped bundles.

Stationary patterns in a discrete bistable reaction-diffusion system:mode analysis

This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure： kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures （kink and pulse） in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.

THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION-DIFFUSION EQUATIONS

The methods of L-p estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boundary values partial derivativeu/partial derivativet = Deltau - lambda \u\(gamma-1) u - betau ((x, t) is an element of Omega x (0, + infinity)), u(x, t) \(partial derivativeOmegax (0, +infinity)) = 0, u(x, 0) = u(0) (x) is an element of H-0(1) (Omega) boolean AND L1+gamma(Omega) (x is an element of Omega). Sufficient and necessary conditions about the extinction of the solutions is given. Here lambda > 0, gamma > 0, beta > 0 are constants, Omega is an element of R-N is bounded with smooth boundary partial derivativeOmega. At last, it is simulated with a higher order equation by using the present methods.

Concentration fluctuation caused by reaction-diffusion coupling near catalytic active sites

The coupling of reaction and diffusion between neighboring active sites in the catalyst pore leads to the spatiotemporal fluctuation in component concentration,which is very importa nt to catalyst performance and hence its optimal design.Molecular dynamics simulation with hard-sphere and pseudo-particle modeling has previously revealed the non-stochastic concentration fluctuation of the reactant/product near isolated active site due to such coupling,using a simple model reaction of A→B in 2D pores.The topic is further developed in this work by studying the concentration fluctuation due to such coupling between neighboring active sites in 3D pores.Two 3D pore models containing an isolated active site and two adjacent active sites were constructed,respectively.For the isolated site,the concentration fluctuation intensifies for larger pores,but the product yield decreases,and for a given pore size,the product yield reaches a peak at a certain reactant concentration.For two neighboring sites,their distance(d)is found to have little effect on the reaction,but significant to the diffusion.For the same reaction competing at both sites,larger d leads to more efficient diffusion and better overall performance.However,for sequential reactions at the two sites,higher overall performance presents at a smaller d.The results should be helpful to the catalyst design and reaction control in the relevant processes.

Global existence and blow-up of solutions to reaction-diffusion system with a weighted nonlocal source

In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior.

Singularly perturbed problem for non-local reaction-diffusion equations involving two small parameters

The problem of two small parameters in ordinary differential equations were extended to that in partial differential equations. The initial boundary problem for the singularly perturbed non-local reaction-diffusion equation was solved. Under suitable conditions, the formal asymptotic solutions were constructed using the method of two-step expansions and the uniform validity of the solutions was proved using the differential inequality.