Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay
The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugatio...
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University of Szeged
2024-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10833 |
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| _version_ | 1846091049101950976 |
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| author | Wenjie Hu Tomas Caraballo |
| author_facet | Wenjie Hu Tomas Caraballo |
| author_sort | Wenjie Hu |
| collection | DOAJ |
| description | The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the almost surely differentiable properties of the RDS, upper estimates of both Hausdorff and fractal dimensions of the random attractors are obtained. |
| format | Article |
| id | doaj-art-e7cbe23d4ca4437abd13f27d5b74f52f |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-e7cbe23d4ca4437abd13f27d5b74f52f2025-01-15T21:24:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-10-0120245612310.14232/ejqtde.2024.1.5610833Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delayWenjie Hu0Tomas CaraballoHunan Normal University, Changsa, Hunan, ChinaThe aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the almost surely differentiable properties of the RDS, upper estimates of both Hausdorff and fractal dimensions of the random attractors are obtained.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10833hausdorff dimensionfractal dimensionrandom dynamical systemrandom attractorsdelaystochastic reaction-diffusion equations |
| spellingShingle | Wenjie Hu Tomas Caraballo Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay Electronic Journal of Qualitative Theory of Differential Equations hausdorff dimension fractal dimension random dynamical system random attractors delay stochastic reaction-diffusion equations |
| title | Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay |
| title_full | Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay |
| title_fullStr | Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay |
| title_full_unstemmed | Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay |
| title_short | Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay |
| title_sort | topological dimensions of random attractors for a stochastic reaction diffusion equation with delay |
| topic | hausdorff dimension fractal dimension random dynamical system random attractors delay stochastic reaction-diffusion equations |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10833 |
| work_keys_str_mv | AT wenjiehu topologicaldimensionsofrandomattractorsforastochasticreactiondiffusionequationwithdelay AT tomascaraballo topologicaldimensionsofrandomattractorsforastochasticreactiondiffusionequationwithdelay |