Asymptotic behavior of travelling fronts of the delayed Fisher equation

This paper is concerned with the travelling wave fronts of the delayed Fisher equation. By Laplace transform, it is proved that all the travelling wave fronts of the delayed Fisher equation satisfy certain asymptotic behavior. Then the existence of such travelling wave fronts is established.     

Global stability of traveling wave fronts for a reaction–diffusion system with a quiescent stage on a one-dimensional spatial lattice

This paper is concerned with the stability of traveling wave fronts for a coupled system of non-local delayed lattice differential equations with a quiescent stage. It shows that under certain conditions all non-critical traveling wave fronts are globally exponentially stable, and critical ftraveling wave fronts are globally algebraically stable by applying the weighted energy method and the semi-discrete Fourier transform.     

Stability of Traveling Wave Fronts for Nonlocal Diffusive Systems

The paper is concerned with stability of traveling wave fronts for nonlocal diffusive systems. We adopt L1-weighted, L1- and L2-energy estimates for the perturbation systems, and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev space.     

Entire solution of the degenerate Fisher equation

This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis and mix. Unfortunately, the entire solutions which behave as two opposite wave fronts with non-critical speed approaching each other from both sides of the X-axis can not be obtained, because the essential difficulty originates from the algebraic decay rate of the fronts with non-critical speed.     

The stability of travelling fronts for general scalar viscous balance law

This paper is concerned with the existence and stability of travelling front solutions for some general scalar viscous balance law. By shooting methods we prove the existence of some class of travelling fronts for any positive viscosity. Further by analytic semigroup theory and detailed spectral analysis, we show that the travelling fronts obtained are asymptotically stable in some appropriate exponentially weighted space. Especially for all sufficiently small viscosity, the travelling waves are proved to be uniformly exponentially stable in the same weighted space.     

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed.     

Generalized fronts in reaction-diffusion equations with bistable nonlinearity

In this paper, we first study the existence of transition fronts (generalized traveling fronts) for reaction-diffusion equations with the spatially heterogeneous bistable nonlinearity. By constructing sub-solution and super-solution we then show that transition fronts are globally exponentially stable for the solutions of the Cauchy problem. Furthermore, we prove that transition fronts are unique up to translation in time by using the monotonicity in time and the exponential decay of such transition fronts.     

Existence,uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model

This paper is concerned with the existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. We first establish the existence of traveling wave fronts by using geometric singular perturbation theory. Then the asymptotic behavior and uniqueness of traveling wave fronts are obtained by using the standard asymptotic theory and sliding method. In addition, our method is also suitable to establish the uniqueness and asymptotic behavior of traveling wave fronts for a cooperative system.     

Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation

This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values.We frst show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infnity.Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts,which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations.Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.     

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.     

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.     

Global stability of traveling wave fronts for non-local delayed lattice differential equations

This paper is concerned with the global stability of traveling wave fronts of a non-local delayed lattice differential equation. By the comparison principle together with the semi-discrete Fourier transform, we prove that, all noncritical traveling wave fronts are globally stable in the form of t^−1/αe^−μt for some constants μ>0 and 0<α≤2, and the critical traveling wave fronts are globally stable in the algebraic form of t^−1/α.     

Reducing the feasible region to a scheduling problem

A special class of scheduling problems is considered. We consider cycle-free sets of fronts correspond to the orderings of a network. If the project is recourse-constrained, the same cycle-free set of fronts can correspond to different orderings. Some cycle-free sets of fronts can be subsets of others. The goal of the paper is to characterize maximal cycle-free sets of fronts because only those are essential for obtaining an optimal schedule.     

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper-lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper-lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov-Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.     

The effective model of a stratified solid-fluid medium as a special case of the Biot model

The Biot model is compared with the effective model of a stratified periodic elastic-fluid medium, and the parameters of the transversally isotropic Biot model that turn it into the effective model are found. In the course of this comparison, some intermediate models, which are generalizations of the effective model and particular cases of the Biot model, are considered. For all of the models, the wave fronts excited by a point source are determined. The distinguishing feature of wave fronts in the intermediate models is the occurrence of double loops on some of them. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 172–195. Translated by L. Molotkov     

Front propagation in periodic excitable media

This paper is devoted to the study of pulsating travelling fronts for reaction‐diffusion‐advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating travelling fronts generalizes that of travelling fronts for planar or shear flows. Various existence, uniqueness and monotonicity results are proved for two classes of reaction terms. Firstly, for a combustion‐type nonlinearity, it is proved that the pulsating travelling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity which arises either in combustion or biological models. Then, the set of possible speeds is a semi‐infinite interval, closed and bounded from below. For each possible speed, there exists a pulsating travelling front which is increasing in time. This result extends the classical Kolmogorov‐Petrovsky‐Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes. © 2002 Wiley Periodicals, Inc.     

Generalized Transition Waves and Their Properties

In this paper, we generalize the usual notions of waves, fronts, and propagation speeds in a very general setting. These new notions, which cover all usual situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces that are parametrized by time. We prove the existence of new such waves for some time‐dependent reaction‐diffusion equations, as well as general intrinsic properties, some monotonicity properties, and some uniqueness results for almost‐planar fronts. The classification results, which are obtained under some appropriate assumptions, show the robustness of our general definitions. © 2012 Wiley Periodicals, Inc.     

Multidimensional stability of traveling fronts in monostable reaction-difusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-difusion equations,including Ginzburg-Landau equations and Fisher-KPP equations.Eckmann and Wayne(1994)showed a one-dimensional stability result of traveling fronts with speeds c c(the critical speed)under complex perturbations.In the present work,we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions(n=2,3),employing weighted energy methods.     

Lie geometry of flat fronts in hyperbolic space

We propose a Lie geometric point of view on flat fronts in hyperbolic space as special Ω-surfaces and discuss the Lie geometric deformation of flat fronts.