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Volume 9, Number 5, 2019, Pages 1769-1800                                                                DOI:10.11948/20180317
Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case
Yanling Meng,Weiguo Zhang,Shengqiang Zhang
Keywords:Entire solutions, upper and lower solutions, traveling wave fronts, comparison principle, upper estimates.
Abstract:
      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.
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