Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 3

Convergence to steady states of parabolic sine-Gordon

Pages: 227 – 248

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n3.a4


Min Gao (University of Chinese Academy of Sciences, Beijing, China; and Innovation Academy for Precision Measurement, Science & Technology, Wuhan Institute of Physics & Mathematics, C.A.S., Wuhan, China)

Jiao Xu (SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China)


Based on the recent surprising work on the symmetry breaking phenomenon of the Allen–Cahn equation [11, 12], we consider the one-dimensional parabolic sine‑Gordon equation with periodic boundary conditions. Particularly, we derive a strong dependence of the non-trivial steady states on the diffusion coefficient $\kappa$ and provide some description on them for $0 \lt \kappa \lt 1$. To further investigate the property of energy associated to the steady states, we give a complete classification and prove the monotonicity of the ground state energy with respect to the diffusion constant $\kappa$. Finally, we identify the exact decay rate of the solution to the parabolic equation together with the explicit leading term for $\kappa \geq 1$.


sine-Gordon equation, steady state, ground state solution, convergence rate, asymptotic behavior

2010 Mathematics Subject Classification

35B10, 35K10, 35K55

Received 7 May 2022

Published 19 May 2023