This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diffusion system. We first give the asymptotic behaviors of time periodic traveling wave solutions at infinity by a dynamical approach coupled with the two-sided Laplace transform. According to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with different speeds propagating from both sides of x-axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.

It is proved that for a given and analytic potential V the nonlinear Schrödinger equation subject to Dirichlet boundary conditions possesses many elliptic invariant tori which carry quasi-periodic motions of high mode.

This paper is concerned with the following Lotka-Volterra competition system with advection in a periodic habitat ∂u1 ∂t = d 1

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This paper is concerned with a time periodic competition-diffusion systemwhere u(t, x) and v(t, x) denote the densities of two competing species, d > 0 is some constant, r i (t), a i (t) and b i (t) are T −periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang [J. Differential Equations 255 (2013), 2402-2435] that this system admits a periodic traveling front connecting two stable semi-trivial T −periodic solutions (p(t), 0) and (0, q(t)) associated to the corresponding kinetic system. Assume further that the wave speed is non-zero, we investigate the asymptotic behavior of the periodic bistable traveling front at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then give some key estimates. Finally, by applying super-and subsolutions technique as well as the comparison principle, we establish the existence and various qualitative properties of entire solutions defined for all time and whole space.