Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation

Abstract

This paper is concerned with the traveling waves and entire solutions for a delayed nonlocal dispersal equation with convolution- type crossing-monostable nonlinearity. We first establish the existence of non-monotone traveling waves. By Ikehara’s Tauberian theorem, we further prove the asymptotic behavior of traveling waves, including monotone and non-monotone ones. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. Finally, the entire solutions are considered. By introducing two auxiliary monostable equations and establishing some comparison arguments for the three equations, some new types of entire solutions are constructed via the traveling wavefronts and spatially independent solutions of the auxiliary equations.

Keywords:

• Nonlocal dispersal
• non-monotone traveling waves
• uniqueness; entire solutions
• crossing-monostable nonlinearity

AMS Subject Classifications:

• 35K57
• 35R20
• 92D25

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The author was supported by NSF of China [grant number 11401478]; Gansu Provincial Natural Science Foundation [grant number 145RJZA220]; SRFDP [grant number 20126203120006].