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http://dx.doi.org/10.4134/JKMS.j170233

GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS WITH A NEW CLASS OF NONLINEARITIES  

Anh, Cung The (Department of Mathematics Hanoi National University of Education)
Tinh, Le Tran (Department of Mathematics Hong Duc University)
Toi, Vu Manh (Faculty of Computer Science and Engineering Thuyloi University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.3, 2018 , pp. 531-551 More about this Journal
Abstract
In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.
Keywords
nonlocal parabolic equation; weak solution; global attractor; fractal dimension; stability; exponential nonlinearity;
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1 S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal. 45 (2005), no. 3-4, 301-312.
2 C.-K. Zhong, M.-H. Yang, and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations 223 (2006), no. 2, 367-399.   DOI
3 D. Li and C. Sun, Attractors for a class of semi-linear degenerate parabolic equations with critical exponent, J. Evol. Equ. 16 (2016), no. 4, 997-1015.   DOI
4 M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987), no. 1-2, 101-147.   DOI
5 S. B. de Menezes, Remarks on weak solutions for a nonlocal parabolic problem, Int. J. Math. Math. Sci. 2006, Art. ID 82654, 10 pp.
6 D. T. Quyet, L. T. Thuy, and N. X. Tu, Semilinear strongly degenerate parabolic equations with a new class of nonlinearities, Vietnam J. Math. 45 (2017), no. 3, 507-517.   DOI
7 C. A. Raposo, M. Sepulveda, O. V. Villagran, D. C. Pereira and, M. L. Santos, Solution and asymptotic behaviour for a nonlocal coupled system of reaction-diffusion, Acta Appl. Math. 102 (2008), 37-56.   DOI
8 J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
9 J. Simsen and J. Ferreira, A global attractor for a nonlocal parabolic problem, Nonlinear Stud. 21 (2014), no. 3, 405-416.
10 R. Temam, Navier-Stokes Equations, revised edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1979.
11 R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
12 M. X. Thao, On the global attractor for a semilinear strongly degenerate parabolic equation, Acta Math. Vietnam. 41 (2016), no. 2, 283-297.   DOI
13 P. T. Thuy and N. M. Tri, Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 1213-1224.   DOI
14 C. T. Anh, Global attractor for a semilinear strongly degenerate parabolic equation on ${\mathbb{R}}^N$, NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 5, 663-678.   DOI
15 C. T. Anh, P. Q. Hung. T. D. Ke, and T. T. Phong, Global attractor for a semilinear parabolic equation involving Grushin operator, Electron. J. Differential Equations 2008 (2008), no. 32, 11 pp.
16 C. T. Anh and T. D. Ke, Existence and continuity of global attractors for a degenerate semilinear parabolic equation, Electron. J. Differential Equations 2009 (2009), no. 61, 13 pp.
17 C. T. Anh and L. T. Tuyet, Strong solutions to a strongly degenerate semilinear parabolic equation, Vietnam J. Math. 41 (2013), no. 2, 217-232.   DOI
18 F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013.
19 T. Caraballo, M. Herrera-Cobos, and P. Marin-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dynam. 84 (2016), no. 1, 35-50.   DOI
20 T. Caraballo, M. Herrera-Cobos, and P. Marin-Rubio, Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal. 121 (2015), 3-18.   DOI
21 M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), no. 7, 4619-4627.   DOI
22 M. Chipot, V. Valente, and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova 110 (2003), 199-220.
23 I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems (Russian), Universitetskie Lektsii po Sovremennoi Matematike., AKTA, Kharkiv, 1999.
24 R. M. P. Almeida, S. N. Antontsev, and J. C. M. Duque, On a nonlocal degenerate parabolic problem, Nonlinear Anal. Real World Appl. 27 (2016), 146-157.   DOI
25 P. G. Geredeli and A. Khanmamedov, Long-time dynamics of the parabolic p-Laplacian equation, Commun. Pure Appl. Anal. 12 (2013), no. 2, 735-754.   DOI
26 A. E. Kogoj and S. Sonner, Attractors for a class of semi-linear degenerate parabolic equations, J. Evol. Equ. 13 (2013), no. 3, 675-691.   DOI
27 A. S. Ackleh and L. Ke, Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3483-3492.   DOI
28 R. M. P. Almeida, S. N. Antontsev, J. C. M. Duque, and J. Ferreira, A reaction-diffusion model for the non-local coupled system: existence, uniqueness, long-time behaviour and localization properties of solutions, IMA J. Appl. Math. 81 (2016), no. 2, 344-364.   DOI
29 M. Anguiano, P. E. Kloeden, and T. Lorenz, Asymptotic behaviour of nonlocal reaction-diffusion equations, Nonlinear Anal. 73 (2010), no. 9, 3044-3057.   DOI
30 P. G. Geredeli, On the existence of regular global attractor for p-Laplacian evolution equation, Appl. Math. Optim. 71 (2015), no. 3, 517-532.   DOI