Attractors for the nonclassical diffusion equations with the driving delay term in time-dependent spaces

Yadan Shi; Yongqin Xie; Ke Li; Zhipiao Tang; Yadan Shi; Yongqin Xie; Ke Li; Zhipiao Tang

doi:10.3934/era.2024320

Electronic Research Archive

2024, Volume 32, Issue 12: 6847-6868. doi: 10.3934/era.2024320

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Research article

Attractors for the nonclassical diffusion equations with the driving delay term in time-dependent spaces

1.
School of General Education, Hunan University of Information Technology, Changsha 410151, China
2.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China

Received: 05 October 2024 Revised: 07 December 2024 Accepted: 19 December 2024 Published: 26 December 2024

In this study, we primarily investigate the asymptotic behavior of solutions associated with a nonclassical diffusion process by memory effects and a perturbed parameter that varies over time. A significant innovation is the consideration of a delay term governed by a function with minimal assumptions: merely measurability and a phase-space that is a time-dependent space of continuously-time-varying functions. By employing a novel analytical approach, we demonstrate the existence and regularity of time-varying pullback $ \mathscr{D} $-attractors. Notably, the nonlinearity $ f $ is unrestricted by any upper limit on its growth rate.
- nonclassical diffusion equation,
- pullback attractors,
- nonlinear delay,
- polynomial growth
Citation: Yadan Shi, Yongqin Xie, Ke Li, Zhipiao Tang. Attractors for the nonclassical diffusion equations with the driving delay term in time-dependent spaces[J]. Electronic Research Archive, 2024, 32(12): 6847-6868. doi: 10.3934/era.2024320

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Abstract

In this study, we primarily investigate the asymptotic behavior of solutions associated with a nonclassical diffusion process by memory effects and a perturbed parameter that varies over time. A significant innovation is the consideration of a delay term governed by a function with minimal assumptions: merely measurability and a phase-space that is a time-dependent space of continuously-time-varying functions. By employing a novel analytical approach, we demonstrate the existence and regularity of time-varying pullback $ \mathscr{D} $-attractors. Notably, the nonlinearity $ f $ is unrestricted by any upper limit on its growth rate.

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