Time-dependent asymptotic behavior of the solution for evolution equation with linear memory

Tingting Liu; Tasneem Mustafa Hussain Sharfi; Qiaozhen Ma; Tingting Liu; Tasneem Mustafa Hussain Sharfi; Qiaozhen Ma

doi:10.3934/math.2023829

AIMS Mathematics

2023, Volume 8, Issue 7: 16208-16227. doi: 10.3934/math.2023829

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Research article

Time-dependent asymptotic behavior of the solution for evolution equation with linear memory

1.
College of Mathematics and Statistics, Northwest Normal University, Gansu, 730070, Lanzhou, China
2.
Faculty of Mathematical science, University of Khartoum, Sudan

Received: 09 February 2023 Revised: 06 April 2023 Accepted: 17 April 2023 Published: 06 May 2023
MSC : 35B40, 35B41, 37L30

In this article, by using the operator decomposition technique, we discuss the existence of a time-dependent global attractor for a nonlinear evolution equation with linear memory within the theory of time-dependent space. Furthermore, the regularity and asymptotic structure of the time-dependent attractor are proved, which means that the time-dependent attractor of the evolution equation converges to the attractor of the limit wave equation when the coefficient $ \varepsilon(t)\rightarrow0 $ as $ t\rightarrow \infty $.
- linear memory,
- time-dependent global attractor,
- regularity,
- asymptotic structure
Citation: Tingting Liu, Tasneem Mustafa Hussain Sharfi, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for evolution equation with linear memory[J]. AIMS Mathematics, 2023, 8(7): 16208-16227. doi: 10.3934/math.2023829

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Abstract

In this article, by using the operator decomposition technique, we discuss the existence of a time-dependent global attractor for a nonlinear evolution equation with linear memory within the theory of time-dependent space. Furthermore, the regularity and asymptotic structure of the time-dependent attractor are proved, which means that the time-dependent attractor of the evolution equation converges to the attractor of the limit wave equation when the coefficient $ \varepsilon(t)\rightarrow0 $ as $ t\rightarrow \infty $.

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