Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

一类非牛顿渗流方程爆破界的估计

本文利用拟线性常微分方程解的非存在性定理得到了一类拟线性反应扩散方程(非牛顿渗流方程)爆破界的估计,从而推广了半线性反应扩散方程(牛顿渗流方程)相应结果.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.     

On Persistence and Stability of Some Spatially Inhomogeneous Systems

Local and global stability and persistence of some coupled map lattices (CMLs) and partial differential equations are studied. A logistic CML with noninteger time step and delay is introduced. The persistence results for reaction-diffusion equations are extendable to the telegraph reaction-diffusion equation for a sufficiently small delay parameter. The stability and persistence results are applied to ecology, physics, economics, and immunology.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.     

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.     

The method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction-diffusion systems

The aim of this paper is to show the existence of solutions of the n-dimensional diffraction problem for weakly coupled quasilinear elliptic reaction-diffusion system. The coefficients of the equations under consideration are allowed to be discontinuous. We extend the method of upper and lower solutions for reaction-diffusion equations with continuous coefficients to the elliptic diffraction problem. An application of these results is given to the steady-state problem of Lotka-Volterra cooperation model with two cooperating species.     

An investigation of chaos in reaction-diffusion equations

The purpose of this article is to investigate graphically and numerically the topic of chaos in reaction-diffusion equations. This article is based on the article by Mitchell and Bruch [1]. One- and two-dimensional forms of the reaction-diffusion equation are discretized using the explicit Euler finite difference scheme. Plots are presented to show the effect of bifurcation parameters on the difference equations. Varying these parameters produce single point, periodic, chaotic, intermittent, and divergent solutions.     

Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. I. Generalized Ginzburg-Landau equations

A group classification of the generalized complex Ginzburg-Landau equations is presented. An approach to group classification of systems of reaction-diffusion equations with general diffusion matrix is formulated.     

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.     

Bifurcations on hemispheres

Summary It is now well known that the number of parameters and symmetries of an equation affects the bifurcation structure of that equation. The bifurcation behavior of reaction-diffusion equations on certain domains with certain boundary conditions isnongeneric in the sense that the bifurcation of steady states in these equations is not what would be expected if one considered only the number of parameters in the equations and the type of symmetries of the equations. This point was made previously in work by Fujii, Mimura, and Nishiura [6] and Armbruster and Dangelmayr [1], who considered reaction-diffusion equations on an interval with Neumann boundary conditions.As was pointed out by Crawford et al. [5], the source of this nongenericity is that reaction-diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains withlarger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction-diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a naturalO(2)-symmetry but may be extended to the full sphere where the natural symmetry group isO(3). We also determine a large class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.     

Global attractivity in monotone and subhomogeneous almost periodic systems

A global attractivity theorem is first proved for a class of skew-product semiflows. Then this result is applied to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations for their global dynamics.     

Perron Theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations

In this paper we revisit the existence of traveling waves for delayed reaction-diffusion equations by the monotone iteration method. We show that Perron Theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction-diffusion systems with time delay. The method is tried out on two classical examples with delay: the predator-prey and Belousov-Zhabotinskii models.     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

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.     

Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity

Entire solutions for monostable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results.     

Positivity and boundedness of solutions of impulsive reaction-diffusion equations

Employing the generalized quasilinearization for nonlinear reaction-diffusion equations, existence of positive bounded solution is proved.