Zhao, Xiaopeng; Liu, Changchun The existence of global attractor for a fourth-order parabolic equation. (English) Zbl 1270.35125 Appl. Anal. 92, No. 1, 44-59 (2013). The authors investigate the following fourth-order parabolic equation \[ u_t +\gamma \Delta^2u-\text{div}\;\left(|\nabla u|^{p-2}\nabla u\right)=0, \] where \(\gamma >0, p>2\) and \(\Omega\) is a two-dimensional smooth bounded domain. This equation is supplemented by the boundary conditions \(\partial u/\partial n=0,\;\partial(\Delta u)/\partial n=0\) on \(\partial \Omega\) and the initial condition \(u(x,0)=u_0\). Taking the initial data in any bounded set \(B\) of \[ U_k=\{u:\;u \in H^2(\Omega),\;|m(u)|\leq k\} \] and using the multipliers \(u\), \( \Delta u\) and \(\Delta^2 u\) as well as \(\Delta^2 u\) (tested against \(\Delta\) applied to the equation), the authors show the boundedness of \(L^2\), \(H^1\), \(H^2\) and \(H^3\) norms (respectively) of \(u\) for \(t\) sufficiently large. These estimates imply the existence of a bounded set in \(H^3(\Omega)\) that is a global attractor in \(U_k\). This results is proved for \(7/2 < p \leq 4\). In the remaining of the paper, the authors prove (under same restriction on \(p\)) the existence of a global attractor in \(H_k\) for \(0 \leq k < 5\) attracting any bounded subset of this space. In this paper, the scheme of the proofs and main results are similar to the ones of: X. Zhao and B. Liu [J. Korean Math. Soc. 49, No. 2, 357–378 (2012; Zbl 1242.35058)]. Reviewer: Jauber C. Oliveira (Florianopolis) Cited in 9 Documents MSC: 35B41 Attractors 35K35 Initial-boundary value problems for higher-order parabolic equations 35K59 Quasilinear parabolic equations Keywords:fourth-order parabolic equation; absorbing set; \(\omega\)-limit set Citations:Zbl 1242.35058 PDFBibTeX XMLCite \textit{X. Zhao} and \textit{C. Liu}, Appl. Anal. 92, No. 1, 44--59 (2013; Zbl 1270.35125) Full Text: DOI References: [1] DOI: 10.1016/S0022-247X(03)00474-8 · Zbl 1029.35110 · doi:10.1016/S0022-247X(03)00474-8 [2] DOI: 10.1016/0022-0248(95)01048-3 · doi:10.1016/0022-0248(95)01048-3 [3] DOI: 10.1017/S0004972700016348 · Zbl 0803.35013 · doi:10.1017/S0004972700016348 [4] DOI: 10.1006/jdeq.1994.1129 · Zbl 0828.35015 · doi:10.1006/jdeq.1994.1129 [5] DOI: 10.1080/00036818608839639 · Zbl 0582.34070 · doi:10.1080/00036818608839639 [6] DOI: 10.1002/cpa.10103 · Zbl 1033.35123 · doi:10.1002/cpa.10103 [7] DOI: 10.1016/j.na.2006.11.005 · Zbl 1128.35048 · doi:10.1016/j.na.2006.11.005 [8] Liu C, Bull. Belg. Math. Soc. Simon Stevin 13 (3) pp 527– (2006) [9] Wu H, Asym. Anal. 51 pp 101– (2007) [10] DOI: 10.1006/jdeq.1998.3429 · Zbl 0912.35029 · doi:10.1006/jdeq.1998.3429 [11] Eden A, Comm. Pure Appl. Anal. 6 (4) pp 1075– (2007) · Zbl 1140.35357 · doi:10.3934/cpaa.2007.6.1075 [12] DOI: 10.4064/ba58-2-3 · Zbl 1206.35049 · doi:10.4064/ba58-2-3 [13] DOI: 10.1007/978-1-4684-0313-8 · doi:10.1007/978-1-4684-0313-8 [14] Ma T, Stability and Bifurcation of Nonlinear Evolution Equations (2006) [15] DOI: 10.1016/j.jmaa.2009.01.035 · Zbl 1173.35028 · doi:10.1016/j.jmaa.2009.01.035 [16] DOI: 10.1016/j.na.2007.03.045 · Zbl 1154.35320 · doi:10.1016/j.na.2007.03.045 [17] DOI: 10.1016/j.na.2009.06.103 · Zbl 1180.35126 · doi:10.1016/j.na.2009.06.103 [18] Grün G, Z. Anal. Anwendungen 20 (4) pp 987– (2001) · Zbl 0996.35026 · doi:10.4171/ZAA/1055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.