Entire solutions of delayed reaction‐diffusion equations

Abstract

This paper is concerned with entire solutions of delayed reaction-diffusion equations. Using the comparing argument and sub-super solutions method, we obtain the existence of entire solutions which behave as two wave fronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables.

Some entire solutions to the competitive reaction diffusion system

“…In fact, in [19] the existence of entire solutions of (1.1) bases on the traveling front solutions which connect (0, 1) and (1, 0), while in [26] it depends on the traveling front solutions which connect (0, 1) and a positive equilibrium. For the existence of entire solutions of systems with or without delays, one can see [17,25,28,29] for more details.…”

“…> 0 and x ∈ R. The reaction terms f and g are quasi-monotone decreasing namely ∂f ∂v of derivatives of the solution to (3.1) are given firstly similarly to[15][16][17]24,[27][28][29][30], which are used to the proof of the existence of entire solutions. For convenience, we briefly show the proof.…”

Entire solutions for a delayed lattice competitive system

“…In addition, they used the exact solutions, which do not satisfy the technical condition, to construct an entire solution for (1.3). For the delayed Lotka-Volterra competition-diffusion system, Lv in [10] obtained the existence of entire solutions by using the comparison principle and super-sub solutions methods, which is similar to that of [12]. Later, for the nonlocal competitiondiffusion system, Li, Zhang and Zhang in [9] have proved the existence of entire solutions to it, in which the asymptotic behavior of the entire solutions was similar to that of [10] and [12].…”

“…For the delayed Lotka-Volterra competition-diffusion system, Lv in [10] obtained the existence of entire solutions by using the comparison principle and super-sub solutions methods, which is similar to that of [12]. Later, for the nonlocal competitiondiffusion system, Li, Zhang and Zhang in [9] have proved the existence of entire solutions to it, in which the asymptotic behavior of the entire solutions was similar to that of [10] and [12]. Very recently, Wang, Liu and Li in [15] investigated the existence of an entire solution for the nonlocal competitive Lokta-Volterra system with delays which extend the results in [9,10,12].…”

Entire solutions for the classical competitive Lotka–Volterra system with diffusion in the weak competition case

“…In [28], Wang and Li showed some sufficient and necessary conditions for this technical condition and partially proved the result still holds without this condition. In addition, except for the above mentioned papers, for the existence of entire solutions of Lotka-Volterra system one can see [18], [20] and [27] for more details. From the above statement, we remark that there is no any results for the existence of entire solutions for system (1.1) in the case (iv) based on the solutions to (1.3) with (1.4) and (1.5).…”