Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

Jianbo Yuan; Shixuan Zhang; Yongqin Xie; Jiangwei Zhang; Jianbo Yuan; Shixuan Zhang; Yongqin Xie; Jiangwei Zhang

doi:10.3934/math.2021684

AIMS Mathematics

2021, Volume 6, Issue 11: 11778-11795. doi: 10.3934/math.2021684

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Research article

Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

1.
School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha, 410114, China
2.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China

Received: 22 February 2021 Accepted: 10 August 2021 Published: 13 August 2021
MSC : 35K57, 35B40, 35B41

In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.
- nonclassical diffusion equation,
- exponential attractor,
- arbitrary polynomial growth,
- fractal dimension,
- global exponentially $ \kappa- $dissipative
Citation: Jianbo Yuan, Shixuan Zhang, Yongqin Xie, Jiangwei Zhang. Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity[J]. AIMS Mathematics, 2021, 6(11): 11778-11795. doi: 10.3934/math.2021684

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Abstract

In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.

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