Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 3

Asymptotic autonomy of kernel sections for Newton–Boussinesq equations on unbounded zonary domains

Pages: 295 – 316

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n3.a4

Authors

Renhai Wang (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Yangrong Li (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Abstract

We study asymptotic autonomy of the kernel sections of an evolution process, which has a forward limiting semigroup. We show that the forward compactness of the kernel sections for the process is a necessary and sufficient condition such that the kernel sections are attracted by the global attractor for the semigroup. The criterion of forward compactness is also established by using the forward-pullback asymptotic compactness of the process. As applications, we obtain nonempty, uniformly bounded and forward compact kernel sections for the non-autonomous Newton–Boussinesq equation defined on an unbounded zonary domain and perturbed by longtime convergent forces. More importantly, the kernel sections are attracted by the global attractor of the autonomous equation.

Keywords

kernel section, pullback attractor, asymptotic autonomy, forward compactness, Newton–Boussinesq equation, unbounded zonary domain

2010 Mathematics Subject Classification

35B40, 35B41, 37L05

Accepted 5 February 2018

Published 30 August 2019