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

Authors

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)

Abstract

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 $\kappa \geq 1$.

Keywords

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