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A periodic problem for the Landau–Ginzburg equation. (English. Russian original) Zbl 1031.35014

Math. Notes 72, No. 2, 204-211 (2002); translation from Mat. Zametki 72, No. 2, 227-235 (2002).
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.

MSC:

35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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