Table of Contents
ISRN Mathematical Analysis
Volume 2013, Article ID 989475, 10 pages
http://dx.doi.org/10.1155/2013/989475
Research Article

Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

1School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha 410004, China
2Hunan Provincial Center for Disease Control and Prevention, Changsha, Hunan 410005, China

Received 31 March 2013; Accepted 24 April 2013

Academic Editors: S. Liu and W. Shen

Copyright © 2013 Yongqin Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. I. L. Bogolubsky, “Some examples of inelastic soliton interaction,” Computer Physics Communications, vol. 13, no. 3, pp. 149–155, 1977. View at Publisher · View at Google Scholar
  2. P. A. Clarkson, R. J. LeVeque, and R. Saxton, “Solitary-wave interactions in elastic rods,” Studies in Applied Mathematics, vol. 75, no. 2, pp. 95–121, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. E. Seyler and D. L. Fanstermacher, “A symmetric regularized long wave equation,” Physics of Fluids, vol. 27, no. 1, pp. 58–66, 1984. View at Google Scholar
  4. W. G. Zhu, “Nonlinear waves in elastic rods,” Acta Solid Mechanica Sinica, vol. 1, no. 2, pp. 247–253, 1980. View at Google Scholar
  5. J. Ferreira and N. A. Larkin, “Global solvability of a mixed problem for a nonlinear hyperbolic-parabolic equation in noncylindrical domains,” Portugaliae Mathematica, vol. 53, no. 4, pp. 381–395, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. W. Zhang and Q. Y. Hu, “Existence and stability of the global weak solution of a nonlinear evolution equation,” Acta Mathematica Scientia A, vol. 24, no. 3, pp. 329–336, 2004. View at Google Scholar · View at MathSciNet
  7. H. A. Levine and J. Serrin, “Global nonexistence theorems for quasilinear evolution equations with dissipation,” Archive for Rational Mechanics and Analysis, vol. 137, no. 4, pp. 341–361, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. A. Levine, P. Pucci, and J. Serrin, “Some remarks on global nonexistence for nonautonomous abstract evolution equations,” in Harmonic Analysis and Nonlinear Differential Equations, vol. 208 of Contemporary Mathematics, pp. 253–263, American Mathematical Society, Providence, RI, USA, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Xie and C. Zhong, “The existence of global attractors for a class nonlinear evolution equation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 54–69, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. N. Carvalho and J. W. Cholewa, “Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time,” Transactions of the American Mathematical Society, vol. 361, no. 5, pp. 2567–2586, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. V. Pata and M. Squassina, “On the strongly damped wave equation,” Communications in Mathematical Physics, vol. 253, no. 3, pp. 511–533, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. V. Pata and S. Zelik, “Smooth attractors for strongly damped wave equations,” Nonlinearity, vol. 19, no. 7, pp. 1495–1506, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. C. Sun and M. Yang, “Dynamics of the nonclassical diffusion equations,” Asymptotic Analysis, vol. 59, no. 1-2, pp. 51–81, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Yang and C. Sun, “Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity,” Transactions of the American Mathematical Society, vol. 361, no. 2, pp. 1069–1101, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Zelik, “Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 921–934, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. P. Fabrie, C. Galusinski, A. Miranville, and S. Zelik, “Uniform exponential attractors for a singularly perturbed damped wave equation,” Discrete and Continuous Dynamical Systems A, vol. 10, no. 1-2, pp. 211–238, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. C. Evans, “Partial differential equation,” GSM 19, American Mathematical Society, Providence, RI, USA, 1998. View at Google Scholar