Entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity

• Published in: Journal of Differential Equations Vol. 265; no. 11; pp. 5520 - 5574
• Main Authors: Wu, Shi-Liang, Chen, Guang-Sheng, Hsu, Cheng-Hsiung
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.12.2018
• Subjects: Entire solution; Epidemic model; Nonlocal dispersal; Traveling wave front; 34K30; Entire solution; 92D30; Nonlocal dispersal; Traveling wave front; 35K57; Epidemic model
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2018.06.012

This paper is concerned with the entire solutions of a nonlocal dispersal epidemic model which arises from the spread of fecally–orally transmitted diseases. Under bistable assumptions, it is well-known that this model has three different types of traveling wave fronts. The annihilating-front and merging-front entire solutions originating from two fronts of the system have also been constructed in [38]. We first prove the uniqueness, Liapunov stability and continuous dependence on shift parameters of annihilating-front entire solutions obtained in [38]. A positive time-derivative estimate for such entire solution is also obtained. Then, we establish the existence of two different types of entire solutions merging three different fronts. Furthermore, we show that these entire solutions are global Lipschitz continuous with respect to the spatial variable x. To the best of our knowledge, it is the first time that the entire solutions originating from three fronts of diffusion systems have been constructed.