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Existence of entire solutions for delayed monostable epidemic models

Authors: Shi-Liang Wu and Cheng-Hsiung Hsu
Journal: Trans. Amer. Math. Soc. 368 (2016), 6033-6062
MSC (2010): Primary 35K57, 35R10; Secondary 35B40, 34K30, 58D25
Published electronically: October 2, 2015
MathSciNet review: 3461026
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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this work is to study the existence of entire solutions for delayed monostable epidemic models with and without the quasi-monotone condition. In the quasi-monotone case, we first establish the comparison principle and construct appropriate sub-solutions and upper estimates. Then the existence and qualitative features of entire solutions are proved by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_{\min }$ and directions and a spatially independent solution, where $ c_{\min }>0$ is the critical wave speed. In the non-quasi-monotone case, some new types of entire solutions are constructed by using the traveling wave fronts and spatially independent solutions of two auxiliary quasi-monotone systems and a comparison theorem for the Cauchy problems of the three systems.

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Additional Information

Shi-Liang Wu
Affiliation: School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China

Cheng-Hsiung Hsu
Affiliation: Department of Mathematics, National Central University, Chungli 32001, Republic of Taiwan

Keywords: Delayed reaction-diffusion system, traveling wave front, entire solution, spatially independent heteroclinic orbit
Received by editor(s): June 15, 2013
Received by editor(s) in revised form: May 12, 2014, and July 25, 2014
Published electronically: October 2, 2015
Additional Notes: The first author’s research was partially supported by the NNSF of China (11301407), NSF of Shaanxi Province (2013JQ1012) and Fundamental Research Funds for the Central Universities (K5051370002)
The second author’s research was supported in part by MST and NCTS of Taiwan
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