One of the variants of the nonlocal Ginzburg–Landau equation is considered. This equation arises in the mathematical modeling of physical phenomena, such as ferromagnetism. For the corresponding initial boundary value problem in the case of periodic boundary conditions, current problems of the theory of infinite-dimensional dynamical systems are considered. The question of the existence of smooth global solutions and a global attractor is studied. Its dimension and structure are determined. All the solutions of the nonlinear initial boundary value problem are found in explicit form. In particular, it was established that all the solutions belonging to the global attractor will be periodic or quasiperiodic functions of the temporal variable.

中文翻译：

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非局部Ginzburg-Landau方程周期边值问题的不变变体

考虑了非局部 Ginzburg-Landau 方程的一种变体。这个方程出现在物理现象的数学建模中，例如铁磁性。对于周期性边界条件下相应的初边值问题，考虑了当前无限维动力系统理论的问题。研究了光滑全局解和全局吸引子的存在性问题。它的尺寸和结构是确定的。非线性初边值问题的所有解都以显式形式求出。特别是，所有属于全局吸引子的解都是时间变量的周期或准周期函数。