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
References:

show all references

References:
 [1] Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241 [2] Mickaël D. Chekroun, Francesco di Plinio, Nathan Glatt-Holtz, Vittorino Pata. Asymptotics of the Coleman-Gurtin model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 351-369. doi: 10.3934/dcdss.2011.4.351 [3] Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067 [4] Jason S. Howell, Irena Lasiecka, Justin T. Webster. Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping. Evolution Equations & Control Theory, 2016, 5 (4) : 567-603. doi: 10.3934/eect.2016020 [5] Marilena N. Poulou, Nikolaos M. Stavrakakis. Global attractor for a Klein-Gordon-Schrodinger type system. Conference Publications, 2007, 2007 (Special) : 844-854. doi: 10.3934/proc.2007.2007.844 [6] Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113 [7] Olivier Goubet, Manal Hussein. Global attractor for the Davey-Stewartson system on $\mathbb R^2$. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1555-1575. doi: 10.3934/cpaa.2009.8.1555 [8] J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625 [9] Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869 [10] Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 [11] Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028 [12] Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193 [13] Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334 [14] Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103 [15] Tamara Fastovska. Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory. Communications on Pure & Applied Analysis, 2007, 6 (1) : 83-101. doi: 10.3934/cpaa.2007.6.83 [16] Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 [17] I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635 [18] Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1 [19] Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure & Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947 [20] Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339

2018 Impact Factor: 1.143