Sign changing periodic solutions for the Chafee–Infante equation

ABSTRACT

This paper is concerned with the existence of sign changing periodic solutions for the Chafee–Infante equation , subject to homogeneous Dirichlet boundary condition. If , where is the m-th eigenvalue of the one-dimensional Laplacian, then there exists a periodic solution, whose zero number is less than or equal to . Specially, if is independent of t, there exists sign changing stationary solutions. Furthermore, numerical simulations verify our result. In some sense, we generalize the result of Bartsch, Polácik and Quittner [Theorem 1.8] 1 to the case of Chafee–Infante equation.

KEYWORDS:

• Sign changing
• periodic solution
• Chafee–Infante equation

AMS SUBJECT CLASSIFICATIONS:

• 35B10
• 35K57
• 35K58

Acknowledgements

The authors would like to express gratitude to Professor Pavol Quittner and Professor Bendong Lou for their generous help and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of H. Huang was supported in part by the Innovation Project of Graduate School of South China Normal University, and the High Level University Developments Innovation Project of South China Normal University [grant number 2016YN12]. The research of R. Huang was supported in part by NSFC [grant number 11671155], [grant number 11371153]; NSFC of Guangdong [grant number 2016A030313418]; and NSFC of Guangzhou [grant number 201607010207], [grant number 201707010136].