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The paper investigates the well-posedness and longtime dynamics of the Kirchhoff wave model with structural nonlinear damping: u t t − ϕ (‖ ∇ u ‖ 2) Δ u + σ (‖ ∇ u ‖ 2) (− Δ) θ u t + f (u) = g (x) , with θ ∈ [ 1 / 2 , 1). We find a new critical exponent p ⁎ ≡ N + 2 N − 2 (> p θ ≡ N + 2 θ N − 2 , N ≥ 3) and show that when the growth exponent p of the nonlinearity f (u) is up to the range: 1 ≤ p < p ⁎ : (i) the IBVP of the equation is well-posed and its solution is of additionally global regularity when t > 0 ; (ii) for each θ ∈ [ 1 / 2 , 1) , the related solution semigroup has in natural energy space H an optimal global attractor A θ whose compactness and attractiveness are in the regularized space H 1 + θ where A θ lies, and an optimal exponential attractor E θ ⁎ whose compactness, boundedness of the fractional dimension and the exponential attractiveness are in H 1 + θ where E θ ⁎ lies, respectively; (iii) the family of global attractors { A θ } θ ∈ [ 1 / 2 , 1) is upper semi-continuous at each point θ 0 ∈ [ 1 / 2 , 1). The paper breaks though the longstanding existed growth restriction: 1 ≤ p ≤ p θ for p θ had been considered a uniqueness index, deepens and extends the results in literature [6,25,27]. [ABSTRACT FROM AUTHOR]

The paper investigates the stability of exponential attractors for a family of semilinear wave equations with gentle dissipation: u t t − Δ u + γ ( − Δ ) α u t + f ( u ) = g ( x ) , with α ∈ ( 0 , 1 / 2 ) . (i) We propose a new criterion on the existence and stability of a family of exponential attractors depending on the perturbation parameters. (ii) By applying this criterion to the equations, we construct a family of exponential attractors A e x p α and show their stability on the dissipative exponent α . [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]