Well-posedness and attractor on the 2D Kirchhoff–Boussinesq models. (July 2020) - Feng, Na; Yang, Zhijian
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).