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The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:utt+Δ2u+(-Δ)αut-Δf(u)=g(x), with α∈(0,1). For general bounded domain Ω⊂RN(N≥1), we show that there exists a critical exponent pα≡N+2(2α-1)(N-2)+ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: 1≤pα, (i) the weak solutions of the equations are of additionally global smoothness when t>0; (i) the related dynamical system possesses a global attractor Aα and an exponential attractor Aαexp in natural energy space for each α∈(0,1), respectively; (ⅲ) the family of global attractors {Aα} is upper semicontinuous at each point α0∈(0,1], i.e., for any neighborhood U of Aα0,Aα⊂U when |α-α0|≪1. These results extend those for structural damping case: α∈[1,2) in [31,32]. [ABSTRACT FROM AUTHOR]

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on RN: utt-φ(x)Δut-φ(x)M(∥∇u∥2)Δu+f(u)=h(x). It proves that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1≤p

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 well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: u t t + Δ 2 u + ( − Δ ) α u t − Δ f ( u ) = g ( x ) , with α ∈ ( 1 , 2 ) . The main results focus on the relations among the dissipative exponent α , the growth exponent p of nonlinearity f ( u ) and the well-posedness and the longtime dynamics of the equations. We find a new critical exponent p α ≡ N + 2 ( 2 α − 1 ) ( N − 2 ( 2 α − 1 ) ) + rather than p ∗ ≡ N + 2 N − 2 ( N ≥ 3 ) as known before and show that when 1 ≤ p < p α : (i) The equations are like parabolic, that is, not only the IBVP of the equations are well-posedness, but also their weak solutions are of higher global regularity as t > 0 . (ii) The related solution semigroup has a global attractor A α in natural energy space, and also has an exponential attractor A e x p α in the sense of partially strong topology. In particular, when 1 ≤ p < p α ′ ≡ N + 2 α ( N − 2 α ) + ( < p α ) , the partially strong topology becomes the strong one. (iii) For any α 0 ∈ [ 1 , 2 ) , the family of global attractors A α is upper semicontinuous at the point α 0 . [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]

The paper studies the existence of global attractor for the generalized double dispersion equation arising in elastic waveguide model u t t − Δ u − Δ u t t + Δ 2 u − Δ u t − Δ g ( u ) = f ( x ) . The main result is concerned with nonlinearities g ( u ) with supercritical growth. In that case we construct a subclass G of the limit solutions and show that it has a weak global attractor. Especially, in non-supercritical case, the weak topology becomes strong, the further regularity of the global attractor is obtained and the exponential attractor is established in natural energy space. [ABSTRACT FROM AUTHOR]

Abstract: The paper studies the global existence, stability and the longtime dynamics of solutions to the initial boundary value problem (IBVP) of an extensible beam equation with nonlinear damping and source terms: . It proves that (i) the IBVP is global well posed provided that either the growth exponent p of the source term is non-supercritical, that is, and if or p is supercritical but is dominated by the growth exponent q of the nonlinear damping , i.e. if ; (ii) the related solution semigroup has in phase space X a finite-dimensional global attractor , which has -regularity, and also an exponential attractor provided either or if ; (iii) especially, when the space dimension , all the above mentioned conclusions hold without any restriction on p and q, that is, . [Copyright &y& Elsevier]

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]