Long-time dynamics of nonlinear MGT-Fourier system

Yang Wang; Jihui Wu; Yang Wang; Jihui Wu

doi:10.3934/math.2024445

AIMS Mathematics

2024, Volume 9, Issue 4: 9152-9163. doi: 10.3934/math.2024445

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Research article

Long-time dynamics of nonlinear MGT-Fourier system

Yang Wang ^,,
Jihui Wu

School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, China

Received: 08 January 2024 Revised: 27 February 2024 Accepted: 28 February 2024 Published: 06 March 2024
MSC : 35B41, 35G61, 35L75, 37L05, 37L30

In this paper, we consider the long-time dynamical behavior of the MGT-Fourier system

$\left\{ {\begin{array}{l} u_{ttt}+\alpha u_{tt}-\beta\Delta u_t-\gamma\Delta u+\eta\Delta\theta+f_1(u,u_t,\theta) = 0,\nonumber\\ \theta_t-\kappa\Delta\theta-\eta\Delta u_{tt}-\eta\alpha\Delta u_t+f_2(u,u_t,\theta) = 0.\nonumber \end{array}} \right. $

First we use the nonlinear semigroup theory to prove the well-posedness of the solutions. Then we establish the existence of smooth finite dimensional global attractors in the system by showing that the solution semigroup is gradient and quasi-stable. Furthermore, we investigate the existence of generalized exponential attractors.
- MGT-Fourier,
- well-posedness,
- global attractors,
- exponential attractors
Citation: Yang Wang, Jihui Wu. Long-time dynamics of nonlinear MGT-Fourier system[J]. AIMS Mathematics, 2024, 9(4): 9152-9163. doi: 10.3934/math.2024445

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Abstract

In this paper, we consider the long-time dynamical behavior of the MGT-Fourier system

$\left\{ {\begin{array}{l} u_{ttt}+\alpha u_{tt}-\beta\Delta u_t-\gamma\Delta u+\eta\Delta\theta+f_1(u,u_t,\theta) = 0,\nonumber\\ \theta_t-\kappa\Delta\theta-\eta\Delta u_{tt}-\eta\alpha\Delta u_t+f_2(u,u_t,\theta) = 0.\nonumber \end{array}} \right. $

First we use the nonlinear semigroup theory to prove the well-posedness of the solutions. Then we establish the existence of smooth finite dimensional global attractors in the system by showing that the solution semigroup is gradient and quasi-stable. Furthermore, we investigate the existence of generalized exponential attractors.

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