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 utt+△2u+ηut-[β(t)+M(∫Ω|▽u(x,t)|2dx)] △u+g(u, t)=f (x,t) Based on a time-uniform priori estimate method, we first in the space H02(Ω) ×L2(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 utt+△2u+ηut-[β(t)+M(∫Ω|▽u(x,t)|2dx)] △u+g(u, t)=f (x,t) Based on a time-uniform priori estimate method, we first in the space H02(Ω) ×L2(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.
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," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 358-362. doi: 10.4236/apm.2012.25052.
References
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[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.