Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

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Abstract

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.

Keywords

Entire solutions Nonlocal dispersal Epidemic model  Traveling wave solutions Asymptotic behavior 

Mathematics Subject Classification

35K57 37C65 92D30 
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Notes

Acknowledgments

Wan-Tong Li: Supported by the NSF of China (11271172). Shi-Liang Wu: Supported by the NSF of China (11301407).

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China

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