Travelling waves for reaction-diffusion equations with time depending nonlinearities

The existence of travelling wave with given end points for parabolic system of nonlinear equations is proven. The nonlinear term depends also on a·x−ct where x is the multidimensional space variable, t—time, c—the speed of the wave and a—the direction of travel.     

Stability of planar waves in reaction-diffusion system

This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t → ∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.     

Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system

The initiation and propagation of reaction-diffusion travellingwaves in two regions coupled together by the linear diffusiveinterchange of the autocatalytic species is considered via aninitial-value problem in which amounts of the autocatalyst areintroduced locally into otherwise uniform concentrations ofthe other species. The reaction in one region is given by quadraticautocatalysis, while the reaction in the other is given by quadraticautocatalysis together with the linear decay of the autocatalyst.A priori bounds for the initial-value problem are obtained first.These, together with the solution valid for small inputs ofthe autocatalyst, enable conditions to be derived under whichtravelling waves can be initiated giving a wave for all withk<2, or if k<(2–1)/(–1), where k and aredimensionless groups corresponding to the rate of chemical decayof the autocatalyst and to the strength of coupling respectively.The global asymptotic stability of the unreacted state is thendiscussed. A solution valid for strong coupling between thetwo regions is then derived. The equations governing the permanent-formtravelling waves are treated in some detail, general propertiesof their solution and a solution valid for weak coupling beingderived. Finally, the large-time solution of the initial-valueproblem is considered. This shows that, when travelling wavesare initiated, they travel with their minimum possible speed:     

Travelling waves in an ionic quadratic autocatalytic chemical system

The influence of a constant electric field on the travelling waves that are initiated in an ionic autocatalytic reaction-diffusion system with quadratic rate law is considered. The necessary conditions for the existence of minimum speed travelling waves are established and examples of such waves have been generated using numerical simulation. The behaviour of the system when the necessary conditions are not satisfied is also examined. This behaviour involves electrophoretic separation and reaction termination, each of which is discussed using examples obtained from numerical simulation.     

Travelling wavefronts of reaction-diffusion equations in cylindrical domains

Abstract

Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This article is aimed at describing refined asymtotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.     

A reaction-diffusion system modelling the spread of bacterial infections

We investigate a system of reaction-diffusion equations which model the spread of a bacterial infection in a human population. A decoupling technique together with global bifurcation theory is used to study the steady-state solutions of the system. The asymptotic behaviour of solutions is discussed by using sub and subersolutions and the quasimonotonicity of the system.     

Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

This paper deals with the existence of travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. Our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system. New iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions. Following this, the existence of travelling wave fronts for reaction-diffusion systems with spatio-temporal delays is established. Finally, we apply the main results to a single-species diffusive model with spatio-temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels.     

Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications

This paper is concerned with the travelling wave fronts of nonlocal reaction-diffusion systems with delays. The existence of travelling wave fronts for nonlocal reaction-diffusion systems with delays is established by using Schauder’s fixed point theorem and upper-lower solution technique. Then these results are applied to the nonlocal delayed Logistic model and the delayed Belousov-Zhabotinskii reaction-diffusion system. Our results show that the time delay can reduce the minimal wave speed while the nonlocality can increase the minimal wave speed. Wan-Tong Li: Supported by NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

Travelling waves in plasma sustained by a laser beam

By means of the Conley connection index theory we prove existence of travelling wave solutions to a system of equations in a two temperature model of laser maintained plasma. These solutions describe locally the motion of plasma boundaries. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Travelling waves in lattice dynamical systems

For a class of one‐dimensional lattice dynamical systems we prove the existence of periodic travelling waves with prescribed speed and arbitrary period. Then we study asymptotic behaviour of such waves for big values of period and show that they converge, in an appropriate topology, to a solitary travelling wave. Copyright © 2000 John Wiley & Sons. Ltd.     

Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control

A two-component reaction-diffusion system modelling a class of spatially structured epidemic systems is considered. The system describes the spatial spread of infectious diseases mediated by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so called zero stabilization. In a series of papers, necessary conditions, and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. The stabilizability with a feedback control of the harvesting type is related to the magnitude of the principal eigenvalue of a certain operator which is not selfadjoint. In this paper, we have proposed an approximating method for this principal eigenvalue. Further, we have faced the problem of finding the optimal position (by translation) of the support of the feedback stabilizing control in order to minimize both the infected population and the pollutant at a certain finite time.     

Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system

We establish the existence of standing waves with one pulse, multiple spikes and transition layers in the nonlinear reaction-diffusion system

u_t=f(u,w)+u_xx,w_t=?²g(u,w)+w_xx,x∈R,     

Travelling waves in dilatant non-Newtonian thin films

We prove the existence of a travelling wave solution for a gravity-driven thin film of a viscous and incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second order partial differential equations for the film's height h and the surfactant's concentration γ are derived by means of lubrication theory applied to the non-Newtonian Navier–Stokes equations.     

Nonmonotone travelling waves in a single species reaction-diffusion equation with delay

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction-diffusion equations , when g∈C²(R₊,R₊) has exactly two fixed points: x₁=0 and x₂=K>0. Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass-type equations with diffusion fall within this subclass of (∗). As an example, we consider the diffusive Nicholson's blowflies equation.     

Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Dynamics of a core-shell reaction-diffusion system

A singular reaction-diffusion system of core-shell type involving a free boundary and an additional dynamical boundary condition is studied. Such systems arise in Chemical Engineering, for example in semibatch regeneration of ion exchangers or in the modelling of desodoration processes. By means of the theory of nonlinear evolution equations, accretive and T-accretive operators as well as Lyapunov functionals it is, in particular, shown that the problem induces a unique nonexpansive, order-preserving semiflow in the natural L¹-setting which converges to a unique steady state.     

Travelling waves of phase transitions in porous media

Liquid–vapour phase changes for a fluid flow through a porous medium are considered; in particular, the model allows for phase mixtures and includes an equilibrium pressure. Existence and uniqueness of travelling waves is established in a wide range of situations; the end states may be formed either by pure phases or mixtures; in the latter case the pressure equals the equilibrium pressure. A formal asymptotic analysis for vanishing relaxation time is carried out to show that the friction and reaction source terms have smoothing effect when the pressure is close to the equilibrium pressure and pure phases are avoided.