# American Institute of Mathematical Sciences

October  2006, 14(4): 801-820. doi: 10.3934/dcds.2006.14.801

## Global attractor for a class of doubly nonlinear abstract evolution equations

 1 Dipartimento di Matematica "F. Casorati", Universitá di Pavia, via Ferrata 1, P.O. Box 27100, Pavia, Italy

Received  December 2004 Revised  August 2005 Published  January 2006

In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space $\H$

$\A(u'(t))+ \B(u(t))-\lambda u(t)$ ∋ $f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$
$u(0)=u_{0},$

where $\A$ is a maximal (possibly multivalued) monotone operator from the Hilbert space $\H$ to itself, while $\B$ is the subdifferential of a proper, convex and lower semicontinuous function φ:$\H\rightarrow (-\infty,+\infty]$ with compact sublevels in $\H$ satisfying a suitable compatibility condition. Finally, $\lambda$ is a positive constant. The existence of solutions is proved by using an approximation-a priori estimates-passage to the limit procedure. The main result of this paper is that the set of all the solutions generates a Generalized Semiflow in the sense of John M. Ball [8] in the phase space given by the domain of the potential φ. This process is shown to be point dissipative and asymptotically compact; moreover the global attractor, which attracts all the trajectories of the system with respect to a metric strictly linked to the constraint imposed on the unknown, is constructed. Applications to some problems involving PDEs are given.

Citation: Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801
 [1] Hernan R. Henriquez. Generalized solutions for the abstract singular Cauchy problem. Communications on Pure and Applied Analysis, 2009, 8 (3) : 955-976. doi: 10.3934/cpaa.2009.8.955 [2] Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047 [3] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 [4] Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149 [5] Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753 [6] Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 [7] Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871 [8] Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039 [9] Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $\mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151 [10] Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 [11] Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114 [12] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [13] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031 [14] Poongodi Rathinasamy, Murugesu Rangasamy, Nirmalkumar Rajendran. Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions. Evolution Equations and Control Theory, 2017, 6 (4) : 599-613. doi: 10.3934/eect.2017030 [15] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [16] Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028 [17] Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic and Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893 [18] Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems and Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317 [19] V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731 [20] Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012

2021 Impact Factor: 1.588