Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T18:49:28.455Z Has data issue: false hasContentIssue false
  • English
  • Français

PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMI-LINEAR DEGENERATE PARABOLIC EQUATION

Published online by Cambridge University Press:  25 August 2010

CUNG THE ANH  and
TANG QUOC BAO
CUNG THE ANH
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam e-mail: [email protected], [email protected]
TANG QUOC BAO
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.

Keywords

35B4135K6535D05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 537 - 554
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Abdelaoui, B. and Peral, I., On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal. 2 (2003), 539–566.CrossRefGoogle Scholar
2.Anh, C. T. and Hung, P. Q., Global existence and long-time behavior of solutions to a class of degenerate parabolic equations, Ann. Pol. Math. 93 (3) (2008), 217–230.CrossRefGoogle Scholar
3.Anh, C. T. and Ke, T. D., Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Anal. 71 (2009), 4415–4422.CrossRefGoogle Scholar
4.Anh, C. T., Chuong, N. M. and Ke, T. D., Global attractor for the m-semiflow generated by a quasiliner degenerate parabolic equation, J. Math. Anal. Appl. 363 (2010), 444–453.CrossRefGoogle Scholar
5.Ball, J. M., On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations, J. Differ. Equ. 27 (1978), 224–265.CrossRefGoogle Scholar
6.Ball, J. M., Continuity properties and global attractor of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502.CrossRefGoogle Scholar
7.Ball, J. M., Global attractor for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), 31–52.CrossRefGoogle Scholar
8.Caldiroli, P. and Musina, R., On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl. 7 (2000), 187–199.CrossRefGoogle Scholar
9.Caraballo, T., Lukasiewicz, G. and Real, J., Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), 484–498.CrossRefGoogle Scholar
10.Caraballo, T., Marin-Rubio, P. and Robinson, J. C., A comparision between to theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal. 11 (2003), 297–322.CrossRefGoogle Scholar
11.Chepyzhov, V. V. and Vishik, M. I., Evolution equations and their trajectory attractor, J. Math. Pures App. 76 (1997), 913–964.CrossRefGoogle Scholar
12.Chepyzhov, V. V. and Vishik, M. I., Attractors for equations of mathematical physics, American Mathematical Society Colloquium, vol. 49 (American Mathematical Society, Providence, RI, 2002).Google Scholar
13.Dautray, R., Lions, J. L., Mathematical analysis and numerical methods for science and technology, Vol. I: Physical origins and classical methods (Springer-Verlag, Berlin, 1985).Google Scholar
14.Karachalios, N. I. and Zographopoulos, N. B., Convergence towards attractors for a degenerate Ginzburg–Landau equation, Z. Angew. Math. Phys. 56 (2005), 11–30.CrossRefGoogle Scholar
15.Karachalios, N. I. and Zographopoulos, N. B., On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differ. Equ. 25 (3) (2006), 361–393.CrossRefGoogle Scholar
16.Li, Y. and Zhong, C. K., Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput. 190 (2007), 1020–1029.Google Scholar
17.Li, Y., Wang, S. and Wu, H., Pullback attractors for non-autonomous reaction-diffusion equations in Lp, Appl. Math. Comput. 207 (2009), 373–379.Google Scholar
18.Lions, J.-L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (Dunod, Paris, 1969).Google Scholar
19.Lu, S. S., Wu, H. Q. and Zhong, C. K., Attractors for nonautonomous 2D Navier–Stokes equations with normal external force, Discrete Contin. Dyn. Syst. 23 (2005), 701–719.CrossRefGoogle Scholar
20.Ma, Q. F., Wang, S. H. and Zhong, C. K., Necessary and sufficient conditions for the existence of global attractor for semigroups and applications, Indian Univ. Math. J. 51 (6) (2002), 1541–1559.CrossRefGoogle Scholar
21.Melnik, V. S., Valero, J., On attractors of multi-valued semiflows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111.CrossRefGoogle Scholar
22.Song, H. T. and Wu, H. Q., Pullback attractors of nonautonomous reaction diffusion equations, J. Math. Anal. Appl. 325 (2007), 1200–1215.CrossRefGoogle Scholar
23.Vishik, M., Asymptotic behaviour of solutions of evolutionary equations (Cambridge University Press, Cambridge, UK, 1992).Google Scholar
24.Zhong, C. K., Yang, M. H. and Sun, C. Y., The existence of global attractors for the norm-to-weak continuous semigoup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ. 15 (2006), 367–399.CrossRefGoogle Scholar
25.Zeidler, E., Nonlinear functional analysis and its applications, vol. II (Springer-Verlag, New York, 1990).Google Scholar