Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

Gaocheng Yue

doi:10.3934/dcdsb.2019101

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2019, Volume 24, Issue 10: 5673-5694. Doi: 10.3934/dcdsb.2019101

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Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

Gaocheng Yue

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received: July 31, 2017

Revised: December 31, 2018

Early access: June 2019

Published: August 2019

The author is supported by NSF of China under Grant 11501289.

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Abstract

In this paper, we study the relations between the long-time dynamical behavior of the perturbed reaction-diffusion equations and the exact reaction-diffusion equations with concave and convex nonlinear terms and prove that bounded sets of solutions of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ t\rightarrow\infty $ and $ \varepsilon\rightarrow 0^+. $ In particular, we show that the trajectory attractor $ \mathscr{U}_ \varepsilon $ of the perturbed reaction-diffusion equations converges to the trajectory attractor $ \mathscr{U}_0 $ of the exact reaction-diffusion equations when $ \varepsilon\rightarrow 0^+. $ Moreover, we derive the upper and lower bounds of the fractal dimension for the global attractor of the perturbed reaction-diffusion equations.

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Mathematics Subject Classification: Primary: 35K57, 37L30, 35B40; Secondary: 35B25.

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