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Abstract The paper investigates longtime dynamics of the Kirchhoff wave equation with strong damping and critical nonlinearities: u t t − (1 + ϵ ‖ ∇ u ‖ 2) Δ u − Δ u t + h (u t) + g (u) = f (x) , with ϵ ∈ [ 0 , 1 ]. The well-posedness and the existence of global and exponential attractors are established, and the stability of the attractors on the perturbation parameter ϵ is proved for the IBVP of the equation provided that both nonlinearities h (s) and g (s) are of critical growth. [ABSTRACT FROM AUTHOR]

The paper investigates the existence of global attractor of the quasi-linear wave equation with strong damping u t t − γ Δ u t − Δ u + f ( u ) = ∇ ⋅ ϕ ′ ( ∇ u ) + g . It proves that the energy solution of the equation is stable and the related dynamical system possesses a (strong) global attractor in natural energy space (rather than a weak one as known before). These results improve the recent ones in [13] . [ABSTRACT FROM AUTHOR]

The paper investigates the global well-posedness and the longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity: u t t − M ( ‖ ∇ u ‖ 2 ) Δ u + ( − Δ ) α u t + f ( u ) = g ( x ) , with α ∈ ( 1 2 , 1 ) . The main results are focussed on the relationships among the growth exponent p of the nonlinearity f ( u ) , the global well-posedness and the longtime dynamics of the equations. We show that (i) even if p is up to the supercritical range, that is, 1 ≤ p < N + 4 α ( N − 4 α ) + , the well-posedness and the longtime behavior of the solutions of the equation are of the characters of the parabolic equation; (ii) when N + 4 α ( N − 4 α ) + ≤ p < N + 4 ( N − 4 ) + , the corresponding subclass G of the limit solutions exists and possesses a weak global attractor. [ABSTRACT FROM AUTHOR]

The paper studies the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on R N : u t t − Δ u t − M ( ‖ ∇ u ‖ 2 ) Δ u + u t + g ( x , u ) = f ( x ) . We establish the well-posedness, the existence of the global and exponential attractors in natural energy space H = H 1 ( R N ) × L 2 ( R N ) in critical nonlinearity case. These results improve not only the recent ones achieved by Yang (2007) [32] , but also ones achieved by Conti, Pata and Squassina (2005) [9] to some extent. [ABSTRACT FROM AUTHOR]

Abstract: The paper studies the longtime dynamics of the damped Boussinesq equation . First, the existence of global solutions to the initial boundary value problem of the equation is obtained provided that the growth exponent of , say , is either non-supercritical (subcritical and critical) or supercritical, especially, the stability of solutions is established when is non-supercritical. Second, the existence of a global attractor and an exponential attractor for the related solution semigroup are respectively established in the non-supercritical case. [Copyright &y& Elsevier]

Abstract: The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation . It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite. [Copyright &y& Elsevier]