Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay

ABSTRACT

This paper deals with the existence of traveling waves for doubly nonlinear degenerate diffusion equations with Nicholson's blowflies-type source term and time delay. We prove the existence of monotone and non-monotone traveling waves by the monotone iteration and Schauder's fixed point theorem respectively. Furthermore, the non-critical traveling waves are proven to be globally $L^1$-stable. The adopted approach for the proof is the weighted energy method combining the compactness analysis.

KEYWORDS:

• Doubly nonlinear degenerate diffusion
• time delay
• traveling waves
• stability

AMS CLASSIFICATIONS:

• 35K57
• 35K59
• 35B35
• 35C07
• 92D25

Acknowledgments

The authors would like to express their sincere thanks for referees' valuable comments and suggestions, which led to the paper a significant improvement.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of R. Huang was supported in part by the NSFC [grant numbers 11971179 and 12126204], Guangdong Basic and Applied Basic Research Foundation [grant numbers 2020A1515010337 and 2020B1515310005]. The research of T. Xu was supported by Guangdong Basic and Applied Basic Research Foundation [grant numbers 2021A1515110951 and 2019A1515010993], China Postdoctoral Science Foundation [grant number 2021M691070], Guangzhou Basic and Applied Basic Research Foundation [grant number 202201010645], NSFC [grant numbers 11971179 and 11871230].