Well-posedness and attractor on the 2D Kirchhoff–Boussinesq models. (July 2020)

• Record Type: Journal Article
• Authors: Feng, Na; Yang, Zhijian
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: Kirchhoff–Boussinesq models -- Well-posedness -- Global attractor -- Exponential attractor; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111803
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The paper studies the well-posedness and the existence of attractors for a class of 2D Kirchhoff–Boussinesq models: u t t + k u t + Δ 2 u = γ d i v { ∇ u 1 + | ∇ u | 2 } + β Δ g ( u ), with β ≥ 0, γ ≥ 0, β + γ > 0 . We show that: (i) the IBVP of the equations is well-posed in natural energy space X 2 and strong solution space X 4, respectively, provided that | g ′ ′ ( s ) | ≤ C ( 1 + | s | 2 ) ; (ii) the related solution semigroup has a global and an (generalized) exponential attractor in X 2 provided that the damping parameter k is suitably large and | g ′ ′ ( s ) | ≤ C ; (iii) in particular when γ = 0, the corresponding Boussinesq model has a subclass J of limit solutions and the subclass J has a weak global attractor in energy space X 1 without any upper bound restriction for the growth exponent of g ( u ) ; (iv) in the cases that either β = 0 or γ = 0, the corresponding model has a global attractor in X 4 provided that | g ′ ′ ( s ) | ≤ C ( 1 + | s | ) and without any restriction for the damping parameter k > 0 . Especially when γ = 0, the corresponding results extend those in Grassell et al. (2009).