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LONG-TIME BEHAVIOR FOR SEMILINEAR DEGENERATE PARABOLIC EQUATIONS ON ℝN

  • Cung, The Anh ;
  • Le, Thi Thuy
  • Received : 2012.03.13
  • Published : 2013.10.31

Abstract

We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

Keywords

semilinear degenerate parabolic equation;weak solution;global attractor;non-compact case;tail estimates method

References

  1. C. T. Anh, N. D. Binh, and L. T. Thuy, On the global attractors for a class of semilinear degenerate parabolic equations, Ann. Polon. Math. 98 (2010), no. 1, 71-89. https://doi.org/10.4064/ap98-1-5
  2. C. T. Anh, N. D. Binh, and L. T. Thuy, Attractors for quasilinear parabolic equations involving weighted p-Laplacian operators, Vietnam J. Math. 38 (2010), no. 3, 261-280.
  3. C. T. Anh, N. M. Chuong, and T. D. Ke, Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation, J. Math. Anal. Appl. 363 (2010), no. 2, 444-453. https://doi.org/10.1016/j.jmaa.2009.09.034
  4. C. T. Anh and P. Q. Hung, Global existence and long-time behavior of solutions to a class of degenerate parabolic equations, Ann. Polon. Math. 93 (2008), no. 3, 217-230. https://doi.org/10.4064/ap93-3-3
  5. C. T. Anh and T. D. Ke, Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Anal. 71 (2009), no. 10, 4415-4422. https://doi.org/10.1016/j.na.2009.02.125
  6. C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Differential Equations Appl. 17 (2010), no. 2, 195-212. https://doi.org/10.1007/s00030-009-0048-3
  7. P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl. 7 (2000), no. 2, 187-199. https://doi.org/10.1007/s000300050004
  8. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.
  9. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. I: Physical origins and classical methods, Springer-Verlag, Berlin, 1985.
  10. N. I. Karachalios and N. B. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys. 56 (2005), no. 1, 11-30. https://doi.org/10.1007/s00033-004-2045-z
  11. N. I. Karachalios and N. B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Anal. 63 (2005), 1749-1768. https://doi.org/10.1016/j.na.2005.03.022
  12. N. I. Karachalios and N. B. Zographopoulos, On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 361-393. https://doi.org/10.1007/s00526-005-0347-4
  13. J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
  14. J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
  15. R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998), no. 1, 71-85. https://doi.org/10.1016/S0362-546X(97)00453-7
  16. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, Philadelphia, 1995.
  17. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, 1997.
  18. B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D 179 (1999), no. 1, 41-52.

Acknowledgement

Supported by : National Foundation for Science and Technology Development(NAFOSTED)