Global Attractor for a Non-Autonomous Beam Equation

ABSTRACT

This work studies the global attractor for the process generated by a non-autonomous beam equation u_{tt}+△^{2}u+ηu_{t}-[β(t)+M(∫_{Ω}|▽u(x,t)|^{2}dx)] △u+g(u, t)=f (x,t) Based on a time-uniform priori estimate method, we first in the space H_{0}^{2}(Ω) ×*L*^{2}(Ω) establish a time-uniform priori estimate of the solution u to the equation, and conclude the existence of bounded absorbing set. When the external term f (x,t) is time-periodic, the continuous semigroup of solution is proved to possess a global attractor.

This work studies the global attractor for the process generated by a non-autonomous beam equation u

Cite this paper

Y. Ren and J. Zhang, "Global Attractor for a Non-Autonomous Beam Equation,"*Advances in Pure Mathematics*, Vol. 2 No. 5, 2012, pp. 358-362. doi: 10.4236/apm.2012.25052.

Y. Ren and J. Zhang, "Global Attractor for a Non-Autonomous Beam Equation,"

References

[1] S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of Applied Mechanics, Vol. 17, 1950, pp. 35-36.

[2] O. F. Ma, S. H. Wang and C. K. Zhong, “Necessary and Sufficient Conditions for the Existence of Global Attractor for Semigroup and Application,” Indiana University Mathematics Journal, Vol. 51, 2002, pp. 529-551.doi.org/10.1512/iumj.2002.51.2255

[3] R. Temam, “Infinite Dimensional Dynamical System in Mechanics and Physical,” 2nd Edition, Spring-Verlag, Nork York, 1997.

[4] Q. Z. Ma and C. K. Zhong, “Existence of Strong Global Attractors for Hyperbolic Equation with Linear Memory,” Applied Mathematics and Computation, Vol. 157, No. 1, 2004, pp. 745-758. doi:10.1016/j.amc.2003.08.080

[5] Q. Z. Ma and C. K. Zhong, “Global Attractors of strong Solutions for Nonclassical Diffusion Equation,” Journal of Lanzhou University, Vol. 40, 2004, pp. 7-9.

[1] S. Woinowsky-Krieger, “The Effect of Axial Force on the Vibration of Hinged Bars,” Journal of Applied Mechanics, Vol. 17, 1950, pp. 35-36.

[2] O. F. Ma, S. H. Wang and C. K. Zhong, “Necessary and Sufficient Conditions for the Existence of Global Attractor for Semigroup and Application,” Indiana University Mathematics Journal, Vol. 51, 2002, pp. 529-551.doi.org/10.1512/iumj.2002.51.2255

[3] R. Temam, “Infinite Dimensional Dynamical System in Mechanics and Physical,” 2nd Edition, Spring-Verlag, Nork York, 1997.

[4] Q. Z. Ma and C. K. Zhong, “Existence of Strong Global Attractors for Hyperbolic Equation with Linear Memory,” Applied Mathematics and Computation, Vol. 157, No. 1, 2004, pp. 745-758. doi:10.1016/j.amc.2003.08.080

[5] Q. Z. Ma and C. K. Zhong, “Global Attractors of strong Solutions for Nonclassical Diffusion Equation,” Journal of Lanzhou University, Vol. 40, 2004, pp. 7-9.