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Asymptotic Behavior of a Nonautonomous $p$ -Laplacian Lattice System

Anhui Gu

School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China

E-mail Address: gahui@swu.edu.cn

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and

Peter E. Kloeden

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

E-mail Address: kloeden@math.uni-frankfurt.de

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https://doi.org/10.1142/S0218127416501741Cited by:40 (Source: Crossref)

Abstract

The existence of a pullback attractor for the nonautonomous $p$ -Laplacian type equations on infinite lattices is established under certain natural dissipative conditions. In particular, there is no restriction on the power index $q$ of the nonlinearity relative to the index $p$ . The forward limiting behavior is also discussed and, under suitable assumptions on the time dependent terms, the lattice system is shown to be asymptotically autonomous with its pullback attractor component sets converging upper semi-continuously to the autonomous global attractor of the limiting autonomous system.

Keywords:

References

Babin, A. & Vishik, M. [1992] Attractors of Evolution Equations (North-Holland, Amsterdam). Google Scholar
Bates, P. W., Lu, K. N. & Wang, B. X. [2001] “ Attractors for lattice dynamical systems,” Int. J. Bifurcation and Chaos 11, 143–153. Link, Web of Science, Google Scholar
Carvalho, A., Cholewa, J. & Dlotko, T. [1999] “ Global attractors for problems with monotone operators,” Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8, 693–706. Google Scholar
Carvalho, A. & Gentile, C. [2003] “ Asymptotic behavior of non-linear parabolic equations with monotone principal part,” J. Math. Anal. Appl. 280, 252–272. Crossref, Web of Science, Google Scholar
Carvalho, A., Langa, J. & Robinson, J. [2012] Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems (Springer, NY). Google Scholar
Chow, S. N. [2003] “ Lattice dynamical systems,” Dynamical Systems, Lecture Notes in Mathematics, Vol. 1822, eds. Macki, J. W. & Zecca, P. (Springer-Verlag, Berlin), pp. 1–102. Crossref, Google Scholar
Damascelli, L. [1998] “ Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,” Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 493–516. Crossref, Web of Science, Google Scholar
Deimling, K. [1977] Differential Equations on Banach Spaces (Springer-Verlag, Heidelberg). Crossref, Google Scholar
Khanmamedov, A. [2006] “ Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain,” J. Math. Anal. Appl. 316, 601–615. Crossref, Web of Science, Google Scholar
Khanmamedov, A. [2009] “ Global attractors for one dimensional $p$ -Laplacian equation,” Nonlin. Anal. 71, 155–171. Crossref, Web of Science, Google Scholar
Kloeden, P. E. & Rasmussen, M. [2011] Nonautonomous Dynamical Systems (American Mathematical Society, Providence). Crossref, Google Scholar
Kloeden, P. E. & Simsen, J. [2015] “ Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents,” J. Math. Anal. Appl. 425, 911–918. Crossref, Web of Science, Google Scholar
Kloeden, P. E. & Lorenz, T. [2016] “ Construction of nonautonomous forward attractors,” Proc. Amer. Math. Soc. 144, 259–268. Crossref, Web of Science, Google Scholar
Kloeden, P. E. & Yang, M. H. [2016] “ Forward attraction in nonautonomous difference equations,” J. Diff. Eqs. Appl. 22, 513–525. Crossref, Web of Science, Google Scholar
Lions, J. [1969] Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires (Gauthier-Villars, Paris). Google Scholar
Temam, R. [1988] Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer, NY). Crossref, Google Scholar
Wang, B. X. [2006] “ Dynamics of systems on infinite lattices,” J. Diff. Eqs. 221, 224–245. Crossref, Web of Science, Google Scholar
Wang, B. X. [2007] “ Asymptotic behavior of non-autonomous lattice systems,” J. Math. Anal. Appl. 331, 121–136. Crossref, Web of Science, Google Scholar
Yang, M. H., Sun, C. Y. & Zhong, C. K. [2007a] “ Global attractors for $p$ -Laplacian equation,” J. Math. Anal. Appl. 337, 1130–1142. Crossref, Web of Science, Google Scholar
Yang, M. H., Sun, C. Y. & Zhong, C. K. [2007b] “ Existence of a global attractor for a $p$ -Laplacian equation in $ℝ^{n}$ ,” Nonlin. Anal. 66, 1–13. Crossref, Web of Science, Google Scholar
Zhou, S. F. & Shi, W. [2006] “ Attractors and dimension of dissipative lattice systems,” J. Diff. Eqs. 224, 172–204. Crossref, Web of Science, Google Scholar
Zhou, S. F. & Han, X. Y. [2012] “ Pullback exponential attractors for non-autonomous lattice systems,” J. Dyn. Diff. Eqs. 24, 601–631. Crossref, Web of Science, Google Scholar