# DYNAMICS OF GLOBAL ATTRACTOR FOR A SEMILINEAR DEGENERATE PARABOLIC EQUATION INVOLVING GRUSHIN OPERATORS

 Title DYNAMICS OF GLOBAL ATTRACTOR FOR A SEMILINEAR DEGENERATE PARABOLIC EQUATION INVOLVING GRUSHIN OPERATORS Publication Type Journal Article Year of Publication 2018 Authors LEE JIHOON Journal Dynamic Systems and Applications Volume 27 Issue 3 Start Page 457 Pagination 18 Date Published 2018 ISSN 1056-2176 AMS Subject Classification 35B05, 35J70 Abstract In this paper we study the dynamics of global attractor of a semilinear parabolic equation involving Grushin operators. First we show that the global attractor is bounded in ${ L^∞(Ω) }$ and ${ D(A) }$. Then we investigate the existence of a Lyapunov function, the injectivity on the global attractor and the squeezing property. Finally, we obtain estimates on upper bound and lower bound of the fractal dimension of the global attractor. PDF https://acadsol.eu/dsa/articles/27/3/2.pdf DOI 10.12732/dsa.v27i3.2 Refereed Designation Refereed Full Text REFERENCES [1] C.T. Anh, Pullback attractor for a non-autonomous parabolic equation involving Grushin operators, Electronic J. Differential Equations (2010), No 11, 1-14. [2] C.T. Anh and T.D. Ke, Existence and continuity of global attractor for a semilinear degenerate parabolic equation, Electronic J. Differential Equations (2009), No 61, 1-13. [3] H. Chen and P. Luo, Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators, Calc. Var. 54 (2015), 2831-2852. [4] L. D’Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2003), 725-734. [5] L. D’Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differential Equations 193 (2003), 511-541. [6] B. Franchi and E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), 1237-1264. [7] V.V. Grushin, A certain class of elliptic pseudo differential operators that are degenerated on a submanifold, Mat. Sb. 84 (1971), 163-195; English translation in: Math. USSR Sbornik 13 (1971), 155-183. [8] A.E. Kogoj and E. Lanconelli, On semilinear ∆λ-Laplace equation, Nonlinear Anal. 75 (2012), 4637-4649. [9] M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987), 101-147. [10] R. Monti and D. Morbidelli, Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J. 131 (2006), 167-202. [11] D.D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc. (JEMS) 12 (2010), 611-654. [12] J.C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2001). [13] R. Temam, Infinite Dimensional Dynamical Systems in Mechanic and Physics, 2nd edition, Springer-Verlag, New York (1997). [14] N.T.C. Thuy and N.M. Tri, Existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operator, Russian J. Math. Phys. 9 (2002), 366-371. [15] Q. Yang, D. Su and Y. Kong, Improved Hardy inequalities for Grushin operators, J. Math. Anal. Appl. 424 (2015), 321-343. [16] X. Yu, Liouville type theorem for nonlinear elliptic equation involving Grushin operators, Commun. Contemp. Math. 17 (2015), 1450050, 12 pp. [17] C.K. Zhong, M.H. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reactiondiffusion equations, J. Differential Equations 223 (2006), 367-399.