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Entire solutions of non-quasi-monotone delayed reaction—diffusion equations with applications

Published online by Cambridge University Press:  03 October 2014

Shi-Liang Wu  and
Cheng-Hsiung Hsu
Shi-Liang Wu
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, People's Republic of China, (slwu@xidian.edu.cn)
Cheng-Hsiung Hsu
Affiliation:
Department of Mathematics, National Central University, Chungli 32001, Taiwan, (chhsu@math.ncu.edu.tw)
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Abstract

We are interested in finding the entire solutions of non-quasi-monotone delayed non-local reaction–diffusion equations. It is well known that the comparison principle is not applicable for such equations. To overcome this difficulty, we introduce two auxiliary quasi-monotone equations and establish some comparison arguments for the three systems. Some new types of entire solutions are then constructed using the comparison argument, the travelling wavefronts and a spatially independent solution of the auxiliary equations. We also extend our arguments to a delayed cellular neural network with non-monotonic output functions and a delayed non-local lattice differential equation with non-monotonic birth functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 1085 - 1112
Copyright
Copyright © Royal Society of Edinburgh 2014 

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