Wenxian Shen

Almost automorphic and almost periodic dynamics in skew-product semiflows

Acknowledgment Abstract Almost automorphy and almost periodicity Skew-product semiflows Applications to differential equations.

Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats

This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices

A lifted lattice is introduced to characterize the stability and hyperbolicity of solutions and to study the existence of topological disorders and chaos in a given coupled map lattice. Time almost-periodic coupled map lattices are also considered through the associated lifted lattices. In particular, the existence of almost-periodic solutions in a discrete-time almost-periodic Nagumo equation is proved.

Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models

Spatial spread and front propagation dynamics is one of the most important dynamical issues in KPP models. Such dynamics of KPP models in time independent or periodic media has been widely studied. Recently, the author of the current paper with Huang established some theoretical foundation for the study of spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. A notion of spreading speed intervals for such models was introduced in the above-mentioned paper and was shown to be the natural extension of the classical concept of the spreading speeds for time independent or periodic KPP models and that it could be used for more general time dependent KPP models. A notion of generalized propagating speed intervals of front solutions and a notion of traveling wave solutions to time almost periodic and space periodic KPP models were also introduced, which are the generalizations of wave speeds and traveling wave solutions in time independent or periodic KPP models. The aim of the current paper is to gain some further qualitative and quantitative understanding of the spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. By applying the principal Lyapunov exponent and the principal Floquet bundle theory for time almost periodic parabolic equations, we provide various useful estimates for spreading and generalized propagating speeds for such KPP models. Under the so-called linear determinacy condition, we show that the spreading speed interval in any given direction is a singleton (called the spreading speed). Moreover, in such a case we establish a variational principle for the spreading speed and prove that there is a front solution of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction. Both the estimates and variational principle provide important and efficient tools for the spreading speeds analysis as well as the spreading speeds computation. Based on the variational principle, the influence of time and space variation of the media on the spreading speeds is also discussed in this paper. It is shown that the time and space variation cannot slow down the spatial spread and that it indeed speeds up the spatial spread except in certain degenerate cases, which provides deep insights into the understanding of the influence of the inhomogeneity of the underline media on the spatial spread in KPP models.

Traveling waves in cellular neural networks

In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

Estimates for the principal spectrum point for certain time-dependent parabolic operators

Non-autonomous parabolic equations are discussed. The periodic case is considered first and an estimate for the principal periodic-parabolic eigenvalue is obtained by relating the original problem to the elliptic one obtained by time-averaging. It is then shown that an analogous bound may be obtained for the principal spectrum point in the almost periodic case. These results have applications to the stability of nonlinear systems and hence, for example, to permanence for biological systems.

On minimal sets of scalar parabolic equations with skew-product structures

Skew-product semi-flow I-t: X x Y -+ X x Y which is generated by Ut -= Uxx + f(Y*t Ix, u, ux), t>0, 0<x< 1, y e Y, D or N boundary conditions is considered, where X is an appropriate subspace of H2(0, 1), (Y, R) is a minimal flow with compact phase space. It is shown that a minimal set E C X x Y of Ilt is an almost 1-1 extension of Y, that is, set Yo = {y E Y I card(E c P 1 (y)) = I} is a residual subset of Y, where P: X x YY is the natural projection. Consequently, if (Y, R) is almost periodic minimal, then any minimal set E C X x Y of Ilt is an almost automorphic minimal set. It is also proved that dynamics of [It is closed in the category of almost automorphy, that is, a minimal set E C X x Y of FIt is almost automorphic minimal if and only if (Y, R) is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.

Spectral theory for random and nonautonomous parabolic equations and applications

We introduce the concept of the principal spectrum for linear forward nonautonomous parabolic partial differential equations. The principal spectrum is a nonempty compact interval. Fundamental properties of the principal spectrum for forward nonautonomous equations are investigated. The paper concludes with applications of the principal spectrum theory to the problem of uniform persistence in some population growth models.

Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations

Abstract The purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones. In the random case, a notion of principal Lyapunov exponent serving as an analog of principal eigenvalue is introduced. It is shown that the principal Lyapunov exponent is deterministic and of simple multiplicity. It is also shown that there is a one-dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign, which serves as an analog of principal eigenfunction. In addition, monotonicity of the principal Lyapunov exponent with respect to the zero-order terms both in the equation and in the boundary condition is proved. When the second- and first-order terms are deterministic, it is proved that the principal Lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time-averaged equation. In the general nonautonomous case, the concepts of principal spectrum, which serves as an analog of principal eigenvalue, and principal Lyapunov exponents are introduced. As is known, the principal spectrum is a compact interval. It is proved in the paper that the principal spectrum contains all the principal Lyapunov exponents. When the second and first-order terms are time independent, a lower estimate of the infimum of the principal spectrum is given in terms of an associated time-averaged equation.

Speeds of Spread and Propagation for KPP Models in Time Almost and Space Periodic Media

The current paper is devoted to the study of spatial spreading and front propagating dynamics of KPP models in inhomogeneous media, particularly, in time almost periodic and space periodic media. While spatial spreading and front propagating dynamics of KPP models in time independent or periodic media has been widely studied, there is little study on such dynamics when the media is nonperiodically inhomogeneous. This paper develops some theoretical foundation for the study of the speeds of spread and propagation for KPP models in time almost periodic and space periodic media. It introduces a notion of spreading speed intervals for such models for the first time, which extends the classical concept of the spreading speeds for time independent or periodic KPP models to time almost periodic models and can be used for more general time dependent models. It also introduces a notion of generalized propagating speed intervals of front solutions to time almost periodic and space periodic KPP models for the first ...