The Global Attractor of Thermoelastic Coupled System

ABSTRACT

In this paper, we consider a class of Sine-Gordon equations which arise from the model of the thermoelastic coupled rod. Firstly, by virtue of the classical semigroup theory, we prove the existence and uniqueness of the mild solution under certain initial-boundary value for above-mentioned equations. Secondly, we obtain the boundedness of solutions by the priori estimates. Lastly, we prove the existence of a global attractor.

In this paper, we consider a class of Sine-Gordon equations which arise from the model of the thermoelastic coupled rod. Firstly, by virtue of the classical semigroup theory, we prove the existence and uniqueness of the mild solution under certain initial-boundary value for above-mentioned equations. Secondly, we obtain the boundedness of solutions by the priori estimates. Lastly, we prove the existence of a global attractor.

Cite this paper

D. Wang and J. Zhang, "The Global Attractor of Thermoelastic Coupled System,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 1 No. 3, 2012, pp. 102-106. doi: 10.4236/ijmnta.2012.13015.

D. Wang and J. Zhang, "The Global Attractor of Thermoelastic Coupled System,"

References

[1] R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1988.

[2] S. Zhu and S. F. Zhou, “Dimension of the Global Attractor for the Damped and Driven SINE-Gordon Equation,” Nonlinear Analysis, Vol. 37, 1999, pp. 389-399.

[3] G. X. Wang and S. Zhu, “Dimension of the Global Attractor for the Discretized Damped Sine-Gordon Equation,” Applied Mathematics and Computation, Vol. 117, No. 2-3, 2001, pp. 257-265. doi:10.1016/S0096-3003(99)00179-4

[4] S. F. Zhou, “Dimension of the Global Attractor for Strongly Damped Nonlinear Wave Equation,” Journal of Mathematical Analysis and Applications, Vol. 233, No. 1, 1999, pp. 102-115. doi:10.1006/jmaa.1999.6269

[5] G. Semion, “Frechet Differentiability for a Damped Sine-Gordon Equation,” Journal of Mathematical Analysis and Applications, Vol. 360, No. 2, 2009, pp. 503-517. doi:10.1016/j.jmaa.2009.06.074

[6] X. Y. Han, “Randon Attractors for Stochastic Sine-Gordon Lattice Systems with Multiplicative White Noise,” Journal of Mathematical Analysis and Applications, Vol. 376, No. 2, 2011, pp. 481-493. doi:10.1016/j.jmaa.2010.11.032

[7] P. Massatt, “Limiting Behavior for Strongly Damped Nonlinear Wave Equations,” Journal of Differential Equations, Vol. 48, No. 3, 1983, pp. 334-349. doi:10.1016/0022-0396(83)90098-0

[1] R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1988.

[2] S. Zhu and S. F. Zhou, “Dimension of the Global Attractor for the Damped and Driven SINE-Gordon Equation,” Nonlinear Analysis, Vol. 37, 1999, pp. 389-399.

[3] G. X. Wang and S. Zhu, “Dimension of the Global Attractor for the Discretized Damped Sine-Gordon Equation,” Applied Mathematics and Computation, Vol. 117, No. 2-3, 2001, pp. 257-265. doi:10.1016/S0096-3003(99)00179-4

[4] S. F. Zhou, “Dimension of the Global Attractor for Strongly Damped Nonlinear Wave Equation,” Journal of Mathematical Analysis and Applications, Vol. 233, No. 1, 1999, pp. 102-115. doi:10.1006/jmaa.1999.6269

[5] G. Semion, “Frechet Differentiability for a Damped Sine-Gordon Equation,” Journal of Mathematical Analysis and Applications, Vol. 360, No. 2, 2009, pp. 503-517. doi:10.1016/j.jmaa.2009.06.074

[6] X. Y. Han, “Randon Attractors for Stochastic Sine-Gordon Lattice Systems with Multiplicative White Noise,” Journal of Mathematical Analysis and Applications, Vol. 376, No. 2, 2011, pp. 481-493. doi:10.1016/j.jmaa.2010.11.032

[7] P. Massatt, “Limiting Behavior for Strongly Damped Nonlinear Wave Equations,” Journal of Differential Equations, Vol. 48, No. 3, 1983, pp. 334-349. doi:10.1016/0022-0396(83)90098-0