Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media

We establish the existence of pulsating type entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time t∈R. By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions.     

具有全局交互作用的时滞周期格微分系统的front-like整体解

研究了一类一维空间周期格上的具有时滞和全局交互作用的微分系统的front⁃like整体解.通过建立适当的比较原理,并融合不同方向的波前解与连接稳定态和不稳定态的空间周期解,构造了front⁃like整体解并证明了一些定性性质.与波前解相比,front⁃like整体解能够展示出新的动力学行为.     

Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case

The paper is concerned with the existence and qualitative features of entire solutions for delayed reaction diffusion monostable systems. Here the entire solutions mean solutions defined on the $ (x,t)\in\mathbb{R}^{N+1} $. We first establish the comparison principles, construct appropriate upper and lower solutions and some upper estimates for the systems with quasimonotone nonlinearities. Then, some new types of entire solutions are constructed by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_* $ and propagation directions and a spatially independent solution, where $c_*>0$ is the critical wave speed. Furthermore, various qualitative properties of entire solutions are investigated. In particularly, the relationship between the entire solution, the traveling wave fronts and a spatially independent solution are considered, respectively. At last, for the nonquasimonotone nonlinearity case, some new types of entire solutions are also investigated by introducing two auxiliary quasimonotone controlled systems and establishing some comparison theorems for Cauchy problems of the three systems.     

Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders

This paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders. Assume that the equation admits three equilibria: two stable equilibria 0 and 1, and an unstable equilibrium θ. It is well known that there are different wave fronts connecting any two of those three equilibria. By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions, we establish three different types of entire solutions. Finally, we analyze a model for shear flows in cylinders to illustrate our main results.     

Spatial Dynamics of Multilayer Cellular Neural Networks

The purpose of this work is to study the spatial dynamics of one-dimensional multilayer cellular neural networks. We first establish the existence of rightward and leftward spreading speeds of the model. Then we show that the spreading speeds coincide with the minimum wave speeds of the traveling wave fronts in the right and left directions. Moreover, we obtain the asymptotic behavior of the traveling wave fronts when the wave speeds are positive and greater than the spreading speeds. According to the asymptotic behavior and using various kinds of comparison theorems, some front-like entire solutions are constructed by combining the rightward and leftward traveling wave fronts with different speeds and a spatially homogeneous solution of the model. Finally, various qualitative features of such entire solutions are investigated.     

Pulsating fronts and front-like entire solutions for a reaction–advection–diffusion competition model in a periodic habitat

This paper is devoted to the study of pulsating fronts and pulsating front-like entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x-axis, which turn out to be unique and Liapunov stable 2-dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2-dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the x-axis, the faster one then invades the slower one as $t\to +\infty$. These kinds of pulsating front-like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat.     

Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks

We obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S¹-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained.     

二维空间格上的具有静止阶段的反应扩散系统的整体解

研究一类二维空间格上的具有静止阶段的反应扩散系统的整体解,这里整体解指的是定义在整个空间和时间上的古典解.构造合适的下解和上估计式,利用比较原理,并利用连接稳定态和不稳定态的空间不依赖解和具有不同波速与传播方向的行波解,证明了整体解的存在性和一些定性性质.     

Generalized fronts in reaction-diffusion equations with mono-stable nonlinearity

We consider transition fronts (generalized traveling fronts) of mono-stable reaction-diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

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.     

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.     

Traveling wave solutions in a delayed diffusive competition system

In this paper we first investigate the existence of traveling wave fronts in a delayed diffusive competition system by constructing a pair of upper and lower solutions. Then we consider the asymptotic behavior of traveling wave solutions at the minus/plus infinity by means of the bilateral Laplace transform. Finally, the monotonicity and uniqueness (up to the translation) of traveling wave solutions are also obtained by the strong comparison principle and the sliding method.     

Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity

In this paper, we prove various qualitative properties of pulsating traveling fronts in periodic media, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov type or general monostable nonlinearities. Besides monotonicity, the main part of the paper is devoted to the exponential behavior of the fronts when they approach their unstable limiting state. In the general monostable case, the logarithmic equivalent of the fronts is shown and for noncritical speeds, the decay rate is the same as in the KPP case. These results also generalize the known results in the homogeneous case or in the case when the equation is invariant by translation along the direction of propagation.     

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.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

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.