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MATHEMATICS

MSC 34K19,35A01,35Q56

Research Article

Invariant manifolds and the global attractor of the generalised nonlocal Ginzburg-Landau equation in the case of homogeneous Dirichlet boundary conditions

A. N. Kulikov, D. A. Kulikov

Demidov Yaroslavl State University, 150003, Yaroslavl, Sovetskaya st. 14, Russia

E-mail: anat_kulikov@mail.ru,kulikov_d_a@mail.ru

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.

Key words: nonlocal Ginzburg-Landau equation, boundary and initial boundary value problems, global solvability, invariant manifolds, global attractors, dimension, structure of global attractors.

DOI: 10.26117/2079-6641-2022-38-1-9-27

Original article submitted: 01.02.2022

Revision submitted: 18.04.2022

For citation. Kulikov A. N., Kulikov D. A. Invariant manifolds and the global attractor of the generalised nonlocal Ginzburg-Landau equation in the case of homogeneous Dirichlet boundary conditions. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 38: 1, 9-27. DOI: 10.26117/2079-6641-2022-38-1-9-27.

Founding. This work was carried out within the framework of a development programme for the Regional Scientific and Educational Mathematical Center of the Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Аgreement on provision of subsidy from the federal budget No. 075-02-2022-886).

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

Acknowledgements. The authors are deeply grateful to the referee for a number of comments that contributed to the improvement of the article.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kulikov A. N., Kulikov D. A., 2022

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Kulikov Anatoliy Nicolaevich — D. Sci. (Phys & Math.), Prof. of Differ. Equation Depart. of Demidov Yaroslavl State University, Yaroslavl, Russia, ORCID 0000-0003-0251-9562.

Kulikov Dmitriy Anatolievich — Ph. D. (Phys & Math.), Associate Professor of Differ. Equation Depart. of Demidov Yaroslavl State University, Yaroslavl, Russia, ORCID 0000-0002-6307-0941.

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