Multiple bifurcations generated by mode interactions in a reaction–diffusion problem

We study multiple bifurcations in a system of reaction–diffusion equations defined on a unit square with Robin boundary conditions. First we investigate linear stabilities of the system at the uniform steady state solution. Then we discuss how multiple bifurcations can be generated by mode interactions of the system, and how these multiple bifurcations can be preserved in the associated discrete system. A continuation-unsymmetric Lanczos algorithm is described to trace discrete solution curves. Numerical experiments on the Brusselator equations are reported.     

Uniform convergence of a weighted average scheme for a nonlinear reaction–diffusion problem

This paper deals with a weighted average scheme (or θ$\theta$-scheme) for solving a nonlinear singularly perturbed parabolic reaction–diffusion problem. The uniform convergence of the weighted average scheme on piecewise uniform and log-meshes is established. Numerical experiments are presented.     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Solvability of reaction–diffusion model with variable exponents

In this article, we study the existence and uniqueness of a weak solution of a degenerate reaction–diffusion parabolic system with variable exponents. This model describes the spread of epidemic diseases with a nonlinear diffusion operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Lyapunov functionals for reaction–diffusion equations with memory

We consider a reaction‐diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

A family of numerical methods for diffusion and reaction–diffusion equations

A family of methods is developed for the numerical solution of second-order parabolic partial differential equations in one space dimension. The methods are second-, third-, or fourth-order accurate in time; five of them are seen to be L₀-stable in the sense of Gourlay and Morris, while the sixth is seen to be A₀-stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction–diffusion problem.     

Global a posteriori error estimates for convection–reaction–diffusion problems

In this note we propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis. Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp. The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs–Poincaré inequality.     

Spike patterns in a reaction–diffusion ODE model with Turing instability

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focus on the model of early carcinogenesis proposed by Marciniak‐Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non‐diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns because of diffusion‐driven instability. To control the accuracy of simulations, we develop a numerical code on the basis of the finite‐element method and adaptive mesh grid. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon on the basis of the emergence of nonstationary structures tending asymptotically to a sum of Dirac deltas. Copyright © 2013 John Wiley & Sons, Ltd.     

A hyperbolic reaction–diffusion model for the hantavirus infection

A hyperbolic reaction–diffusion model for the hantavirus infection, generalizing the parabolic set of equations recently derived by Abramson and Kenkre, is proposed within the context of Extended Thermodynamics. The model, as in the parabolic case, captures some of the realistic features of the dynamics of hantavirus in mice population, while it avoids the unphysical features concerning the instantaneous diffusive effects typical of parabolic equations. Traveling wave solutions, related to the spread of the infection in the landscape, are investigated. Both analytical and numerical results obtained herein are discussed and validated from the behavior of the biological system. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and asymptotic behaviour in a reaction–diffusion system

This paper is concerned with some qualitative analysis for a coupled system of five reaction–diffusion equations which arises from a physiology model. The uniform boundedness of the time-dependent solution is obtained under various boundary conditions. Sufficient conditions are also given to ensure the asymptotic stability of the non-negative steady-state solutions under Dirichlet or Robin boundary condition for each component. Under homogeneous Neumann boundary condition for some components the time-dependent solution is proven to converge to a constant steady state determined by the initial functions.     

Global boundedness of solutions to a reaction–diffusion system

We obtain in this paper the global boundedness of solutions to a Fujita‐type reaction–diffusion system. This global boundedness results from diffusion effect, homogeneous Dirichlet boundary value conditions and appropriate reactions. Copyright © 1999 John Wiley & Sons, Ltd.     

A Lyapunov functional for a triangular reaction–diffusion system with nonlinearities of exponential growth

The aim of this work is to study the global existence of solutions to a triangular system of reaction–diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth. Copyright © 2012 John Wiley & Sons, Ltd.     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Null-controllability of some reaction–diffusion systems with one control force

This work is concerned with the null-controllability of semilinear parabolic systems by a single control force acting on a subdomain.     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

The streamline–diffusion method for a convection–diffusion problem with a point source

A linear singularly perturbed convection–diffusion problem with a point source is considered. The problem is solved using the streamline–diffusion finite element method on a class of Shishkin–type meshes. We prove that the method is almost optimal with uniform second order of convergence in the maximum norm. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.     

Global existence and blow-up to a degenerate reaction–diffusion system with nonlinear memory

In this paper, we consider a degenerate reaction–diffusion system coupled by nonlinear memory. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Furthermore, the blow-up rates are obtained.     

The Decay of mass for a nonlinear fractional reaction–diffusion equation

We study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ?_tu = ? t^σ( ? Δ)^{α ∕ 2}u ? h(t)u^p (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows:

• if , then there exists M _∞ ∈ (0, ∞ ) such that ;
• if , then .

Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with time-varying delays

The stability property of reaction–diffusion generalized Cohen–Grossberg neural networks (GDCGNNs) with time-varying delay are considered. Without assuming the monotonicity and differentiability of activation functions, nor symmetry of synaptic interconnection weights, delay independent and easily verifiable sufficient conditions to guarantee the exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, by employing the method of variational parameter and inequality technique. One example is given to illustrate the theoretical results.