The stability of traveling waves for Allen–Cahn equations with fractional Laplacian

Abstract

In this paper, we mainly consider the stability of traveling waves for the Allen–Cahn equation with fractional Laplacian. By the comparison principle and the perturbation theory, we can locate the spectrum of the linearized operator. Using the spectral analysis of the linearized operator, we obtain that the traveling waves are exponentially asymptotically stable.

COMMUNICATED BY:

• Alexander Pankov

Keywords:

• Traveling waves
• the stability
• Allen-Cahn equation
• fractional Laplacian
• spectral analysis

MSC 2010:

• 35C07
• 47G10
• 47D06

Disclosure statement

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

Additional information

Funding

The work is supported by National Natural Science Foundation of China [grant number 11701341] and Natural Science Foundation of Shandong Province [grant number ZR2017BA005].