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Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where 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 (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood–Paley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting. [Copyright &y& Elsevier]