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Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 31--41 | DOI:10.5890/DNC.2021.03.003

Jishan Fan$^1$, Gen Nakamura$^2$, and Mei-Qin Zhan$^3$

$^1$ Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

$^2$ Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

$^3$ Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL32224, USA

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Abstract

In this paper, we prove the uniqueness of weak solutions $(f, Q)$ to the phase-lock equations with $f_0 \in L^2$ and $Q_0 \in L^3$ when the space dimension $d = 3.$ We also prove the uniqueness of weak solutions $(f, a)$ to the Ginzburg-Landau equations with $(f_0, a_0) \in L^p \times L^p$ and $1 < p < 2$ when $d = 1.$ We will also present a result on the decay of $Q$ as time $t\to\infty.$

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