Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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GLOBAL ATTRACTOR OF THE WEAKLY DAMPED WAVE EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

  • Zhu, Chaosheng (School of mathematics and statistics Southwest University)
  • Received : 2009.07.25
  • Published : 2012.01.31
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Abstract

In this paper, the main purpose is to study existence of the global attractors for the weakly damped wave equation with nonlinear boundary conditions. To this end, we first show that the existence o a bounded absorbing set by the perturbed energy method. Secondly, we utilize the decomposition of the solution operator to verify the asymptotic compactness.

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References

  1. R. B. Anibal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data, J. Differential Equations 181 (2002), no. 1, 165-196. https://doi.org/10.1006/jdeq.2001.4072
  2. A. N. Carvalho, S. M. Oliva, A. L. Pereira, and R. B. Anibal, Attractors for parabolic problems with nonlinear boundary conditions, J. Math. Anal. Appl. 207 (1997), no. 2, 409-461. https://doi.org/10.1006/jmaa.1997.5282
  3. G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9) 58 (1979), no. 3, 249-273.
  4. V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9) 69 (1990), no. 1, 33-54.
  5. I. N. Kostin, Long-time behavior of solutions to a semilinear wave equation with boundary damping, Dynam. Control 11 (2001), no. 4, 371-388. https://doi.org/10.1023/A:1020819205328
  6. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), no. 2, 163-182. https://doi.org/10.1016/0022-0396(83)90073-6
  7. I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79 (1989), no. 2, 340-381. https://doi.org/10.1016/0022-0396(89)90107-1
  8. I. Lasiecka and R. Triggiani, Uniform Stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim. 25 (1992), no. 2, 189-224. https://doi.org/10.1007/BF01182480
  9. J. E. Munoz Rivera, Smoothness effect and decay on a class of nonlinear evolution equation, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 2, 237-260. https://doi.org/10.5802/afst.747
  10. D. Tataru, Uniform decay rates and attractors for evolution PDE's with boundary dissipation, J. Differential Equations 121 (1995), no. 1, 1-27. https://doi.org/10.1006/jdeq.1995.1119
  11. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York, Springer, 1988.
  12. E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal. 1 (1988), no. 2, 161-185.