Geodesic
Invariant manifolds and the global attractor of the generalised nonlocal Ginzburg-Landau equation in the case of homogeneous dirichlet boundary conditions
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 38 (2022) no. 1, pp. 9-27

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Two versions of the generalized nonlocal Ginzburg-Landau equation are considered. Both of these options are studied together with the homogeneous Dirichlet boundary conditions. For the corresponding initial-boundary value problems, the existence of solutions is shown for all positive values of the evolution variable. For solutions to initial-boundary value problems, explicit formulas are obtained in the form of Fourier series. The properties of solutions of the corresponding initial-boundary value problems are studied. In the second part of the work, the question of the existence of global attractors for solutions to the studied boundary value problems is considered. The question of the properties of global attractors is studied. In particular, an answer is given about the Euclidean dimension of such attractors. Sufficient conditions are given under which the global attractor will be finite-dimensional. A variant of the nonlocal Ginzburg-Landau equation is distinguished, when the global attractor is infinite-dimensional.
Keywords: nonlocal Ginzburg-Landau equation, boundary and initial boundary value problems, global solvability, invariant manifolds, global attractors, structure of global attractors.
Mots-clés : dimension 
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     title = {Invariant manifolds and the global attractor of the generalised nonlocal {Ginzburg-Landau} equation in the case of homogeneous dirichlet boundary conditions},
     journal = {Vestnik KRAUNC. Fiziko-matemati\v{c}eskie nauki},
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     url = {http://geodesic.mathdoc.fr/item/VKAM_2022_38_1_a0/}
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A. N. Kulikov; D. A. Kulikov. Invariant manifolds and the global attractor of the generalised nonlocal Ginzburg-Landau equation in the case of homogeneous dirichlet boundary conditions. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 38 (2022) no. 1, pp. 9-27. http://geodesic.mathdoc.fr/item/VKAM_2022_38_1_a0/