Coupled non-local periodic parabolic systems with time delays

Abstract

In this article, we study the comparison principle and long-time behaviour of the solutions for a coupled system of semilinear periodic parabolic equations with discrete time delays and non-local boundary conditions. A new comparison principle is obtained for such a coupled system, which extends some existing results. Moreover, it is shown that for a given sector 𝒥₀ ≔ {u; û ≤ u ≤ ũ}, which is generated by the given upper and lower solutions û and ũ of the corresponding non-local periodic parabolic system, the non-local periodic parabolic system has a maximal periodic solution and a minimal periodic solution u _T such that the ω–limit set of any trajectory of the delayed parabolic system in 𝒥₀ belongs to the sector 𝒥 _T ≔ {u; u _T ≤ u ≤ }. Applications to a Lotka-Volterra cooperating model in ecology are given.

Keywords:

• coupled parabolic system
• non-local boundary conditions
• comparison principle
• asymptotic behavior
• time delays

Acknowledgements

The authors would like to thank the referee very much for his/her very careful reading and very valuable comments and suggestions. The authors would also like to thank Prof. C.V. Pao sincerely for sending us the references 9,10, which are very helpful for us. R.-N. Wang acknowledges support from the NSF of Jiangxi Province, China (2007GQS2056). T.-J. Xiao and J. Liang acknowledge support from the NSF of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, and the NCET-04-0572.