Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities

Q-conditional symmetries (nonclassical symmetries) for the general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reaction-diffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.     

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions. Received December 21, 2000; accepted August 30, 2001.     

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.     

Existence and uniqueness of solutions of infinite systems of semilinear parabolic differential-functional equations in arbitrary domains in ordered banach spaces

We consider the Fourier first initial-boundary value problem for a weakly coupled infinite system of semilinear parabolic differential-functional equations of reaction-diffusion type in arbitrary (bounded or unbounded) domain. The right-hand sides of the system are functionals of unknown functions of the Volterra type. Differential-integral equations give examples of such equations. To prove the existence and uniqueness of the solutions, we apply the monotone iterative method. The underlying monotone iterative scheme can be used for the computation of numerical solution.     

具有无穷时滞的Volterra反应扩散方程组正解与有界正解的存在唯一性

本文利用Ｃ-ｈ空间理论,ＢＣ_ｈ空间理论和上下解方法研究具有无穷时滞的 Ｖｏｌｔｅｒｒａ反应扩散方程组正解和有界正解的存在唯一性,并给出有关应用的例子．     

Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives

This paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time. The Fujita exponent is determined. Then, these results are extended to systems of reaction-diffusion equations. Our new results shed lights on important practical questions.     

Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton-Krylov method

The matrix-free Newton-Krylov method that uses the GMRES algorithm (an iterative algorithm for solving systems of linear algebraic equations) is used for the parametric continuation of the solitary traveling pulse solution in a three-component reaction-diffusion system. Using the results of integration on a short time interval, we replace the original system of nonlinear algebraic equations by another system that has more convenient (from the viewpoint of the spectral properties of the GMRES algorithm) Jacobi matrix. The proposed parametric continuation proved to be efficient for large-scale problems, and it made it possible to thoroughly examine the dependence of localized solutions on a parameter of the model.     

Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications

The paper considers the existence and uniqueness of pseudo almost periodic (mild) solutions to some classes of first-order partial neutral functional-differential equations. Upon making some suitable assumptions, existence and uniqueness results are obtained. Applications include both a partial integro-differential equation arising in control systems and the scalar reaction-diffusion equation with delay.     

Planar Radial Spots in a Three-Component FitzHugh�CNagumo System

Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary spots described here.     

Traveling waves in excitable media

We consider a two-component reaction-diffusion model that describes the oxygenation of CO molecules on the surface of platinum in the one-dimensional case. The partial differential equations of the model are reduced to a system of ordinary differential equations. We show that the system of partial differential equations with fixed parameter values has a family of autowave solutions running along the spatial axis at various velocities. These solutions are described by some singular attractors and limit cycles of the corresponding period in the system of ordinary differential equations.     

The uniqueness of solutions to discrete, vector, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.     

Diffusion effect and stability analysis of a predator-prey system described by a delayed reaction-diffusion equations

In this paper, we consider a delayed reaction-diffusion equations which describes a two-species predator-prey system with diffusion terms and stage structure. By using the linearization method and the method of upper and lower solutions, we study the local and global stability of the constant equilibria, respectively. The results show that the free diffusion of the delayed reaction-diffusion equations has no effect on the populations when the diffusion is too slow; otherwise, the free diffusion has a certain influence on the populations, however, the influence can be eliminated by improving the parameters to satisfy some suitable conditions.     

Spectral methods for some singularly perturbed third order ordinary differential equations

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.      

On the Structure of Spectra of Modulated Travelling Waves

Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses o fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the esolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appopriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floe index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.     

Motion of curves and solutions of two multi-component mKdV equations

Two classes of multi-component mKdV equations have been shown to be integrable. One class called the multi-component geometric mKdV equation is exactly the system for curvatures of curves when the motion of the curves is governed by the mKdV flow. In this paper, exact solutions including solitary wave solutions of the two- and three-component mKdV equations are obtained, the symmetry reductions of the two-component geometric mKdV equation to ODE systems corresponding to it’s Lie point symmetry groups are also given. Curves and their behavior corresponding to solitary wave solutions of the two-component geometric mKdV equation are presented.     

Complex nonlinear dynamics in subdiffusive activator-inhibitor systems

In this article we analyze the linear stability of nonlinear time-fractional reaction-diffusion systems. As an example, the reaction-subdiffusion model with cubic nonlinearity is considered. By linear stability analysis and computer simulation, it was shown that fractional derivative orders can change substantially an eigenvalue spectrum and significantly enrich nonlinear system dynamics. A overall picture of nonlinear solutions in subdiffusive reaction-diffusion systems is presented.     

Numerical solution of two-dimensional reaction-diffusion Brusselator system

In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction-diffusion Brusselator system. The system arises in the modeling of certain chemical reaction-diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.     

Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a non-Newtonian filtration system

The prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first. Then the result of nonexistence for a differential equation system of radially nonincreasing positive solutions is implied. By using this nonexistence result, blow-up estimates for a class of quasilinear reaction-diffusion systems (non-Newtonian filtration systems) are established to extend the result of semilinear reaction-diffusion (Fujita type) systems.     

Numerical Method for Homoclinic and Heteroclinic Orbits of Neuron Models

A twisted heteroclinic cycle was proved to exist more than twenty- five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their traveling wave moving frame. The result implies the existence of infinitely many traveling front waves and infinitely many traveling back waves for the system. However efforts to numerically render the twisted cycle were not fruit- ful for the main reason that such orbits are structurally unstable. Presented here is a bisectional search method for the primary types of traveling wave solu- tions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo equations represent. The algorithm converges at a geometric rate and the wave speed can be approximated to significant precision in principle. The method is then applied for a recently obtained axon model with the conclusion that twisted heteroclinic cycle maybe more of a theoretical artifact.     

Comparison Results for a Kind of Impulsive Parabolic Equations with Application to Population Dynamics

In this paper,for a semi-linear parabolic partial differential equations with impulsive effects,theexistence-comparison theorem and comparison principles are established using the method of upper and lowersolutions.These results are applied to obtain the stability results of the steady-state solutions in a reaction-diffusion equations modelling two competing species with instantaneous stocking.