International Journal of Differential Equations

Attractors for Nonautonomous Parabolic Equations without Uniqueness

Cung The Anh and Nguyen Dinh Binh

Full-text: Open access

Abstract

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

Article information

Source
Int. J. Differ. Equ., Volume 2010 (2010), Article ID 103510, 17 pages.

Dates
Received: 9 September 2009
Revised: 12 February 2010
Accepted: 8 April 2010
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399922

Digital Object Identifier
doi:10.1155/2010/103510

Mathematical Reviews number (MathSciNet)
MR2652393

Zentralblatt MATH identifier
1239.35024

Citation

The Anh, Cung; Dinh Binh, Nguyen. Attractors for Nonautonomous Parabolic Equations without Uniqueness. Int. J. Differ. Equ. 2010 (2010), Article ID 103510, 17 pages. doi:10.1155/2010/103510. https://projecteuclid.org/euclid.ijde/1485399922


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References

  • V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2002.MR1868930
  • R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68, Springer, New York, NY, USA, 2nd edition, 1997.MR1441312
  • J. M. Ball, “Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,” Journal of Nonlinear Science, vol. 7, no. 5, pp. 475–502, 1997.MR1462276
  • V. V. Chepyzhov and M. I. Vishik, “Evolution equations and their trajectory attractors,” Journal de Mathématiques Pures et Appliquées, vol. 76, no. 10, pp. 913–964, 1997.MR1489945
  • V. S. Melnik and J. Valero, “On attractors of multivalued semi-flows and differential inclusions,” Set-Valued Analysis, vol. 6, no. 1, pp. 83–111, 1998.MR1631081
  • V. S. Melnik and J. Valero, “On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions,” Set-Valued Analysis, vol. 8, no. 4, pp. 375–403, 2000.MR1802241
  • A. V. Kapustyan, “Global attractors of a nonautonomous reaction-diffusion equation,” Differential Equations, vol. 38, no. 10, pp. 1467–1471, 2002, translation from Differensial'nyeUravneniya, vol. 38, no. 10, pp. 1378–1381, 2002.MR1984456
  • O. V. Kapustyan and D. V. Shkundīn, “The global attractor of a nonlinear parabolic equation,” Ukrainian Mathematical Journal, vol. 55, no. 4, pp. 446–455, 2003.MR2072547
  • J. Valero and A. Kapustyan, “On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 614–633, 2006.MR2262232
  • A. V. Kapustyan, V. S. Melnik, and J. Valero, “Attractors of multivalued dynamical processes generated by phase-field equations,” International Journal of Bifurcation and Chaos, vol. 13, no. 7, pp. 1969–1983, 2003.MR2015645
  • J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete and Continuous Dynamical Systems. Series A, vol. 10, no. 1-2, pp. 31–52, 2004.MR2026182
  • G. R. Sell, “Global attractors for the three-dimensional Navier-Stokes equations,” Journal of Dynamics and Differential Equations, vol. 8, no. 1, pp. 1–33, 1996.MR1388163
  • T. Kaminogo, “Kneser families in infinite-dimensional spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 5, pp. 613–627, 2001.MR1838950
  • J. Valero, “On the Kneser property for some parabolic problems,” Topology and Its Applications, vol. 153, no. 5-6, pp. 975–989, 2005.MR2201479
  • A. V. Kapustyan and J. Valero, “On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 254–272, 2009.MR2526826
  • R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume I: Physical Origins and Classical Methods, Springer, Berlin, Germany, 1985.
  • N. I. Karachalios and N. B. Zographopoulos, “Convergence towards attractors for a degenerate Ginzburg-Landau equation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 1, pp. 11–30, 2005.MR2112838
  • N. I. Karachalios and N. B. Zographopoulos, “On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 3, pp. 361–393, 2006.MR2201677
  • C. T. Anh and P. Q. Hung, “Global existence and long-time behavior of solutions to a class of degenerate parabolic equations,” Annales Polonici Mathematici, vol. 93, no. 3, pp. 217–230, 2008.MR2403656
  • C. T. Anh, N. D. Binh, and L. T. Thuy, “On the global attractors for a class of semilinear degenerate parabolic equations,” Annales Polonici Mathematici, vol. 98, no. 1, pp. 71–89, 2010.
  • P. Caldiroli and R. Musina, “On a variational degenerate elliptic problem,” Nonlinear Differential Equations and Applications, vol. 7, no. 2, pp. 187–199, 2000.MR1771466
  • J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Paris, France, 1969.