The stability of the equilibria of the Allen–Cahn equation with fractional diffusion

ABSTRACT

This paper is mainly concerned with the stability of the equilibria of the Allen–Cahn equation where the standard Laplacian diffusion is replaced by the fractional Laplacian diffusion. We shall show that the state $u\equiv 0$ is stable with respect to some perturbations which are not too large on a set, but is unstable with respect to some perturbations with bounded support set. We use the comparison principle to obtain the results.

KEYWORDS:

• Fractional diffusion
• stability
• Allen–Cahn equation
• the comparison principle

AMS SUBJECT CLASSIFICATIONS:

• 35B35
• 35C07
• 35K91

Acknowledgements

The authors would like to thank the anonymous referee for their careful reading and valuable comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by The Natural Science Foundation of China [grant number 11701341].