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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 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 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]