# American Institute of Mathematical Sciences

September  2017, 37(9): 4729-4751. doi: 10.3934/dcds.2017203

## On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors

 Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  December 2016 Revised  April 2017 Published  June 2017

A semilinear Timoshenko-Coleman-Gurtin system is studied. The system describes a Timoshenko beam coupled with a temperature with Coleman-Gurtin law. Under some assumptions on nonlinear damping terms and nonlinear source terms, we establish the global well-posedness of the system. The main result is the long-time dynamics of the system. By using the methods developed by Chueshov and Lasiecka, we get the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension. Result on exponential attractors of the system is also proved.

Citation: Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203
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##### References:
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