A periodic problem for the Landau--Ginzburg equation (Q1810180)

This paper is devoted to the following periodic problem for the complex Landau-Ginzburg equation \[ \begin{gathered} u_t+ \lambda|u|^2 u+ au-(\alpha+ i\beta)\Delta u= 0,\quad\alpha\in \Omega,\quad t>0,\\ u|_{t=0}=\overline u(x),\end{gathered}\tag{1} \] where \(x= (x_1,\dots, x_n)\), \(n\geq 1\), \(\Omega\) is the \(n\)-dimensional cube with edge length \(2\pi\), \(a\), \(\alpha\), \(\beta\) are real numbers, and \(\lambda\) is a complex number. The authors show that in the case of small initial data there exists a unique classical solution of (1), and an asymptotics of this solution uniform in the space variable is given. It is shown that, the leading term of the asymptotics decreasing exponential in time.