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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 longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model . It proves that when the space dimension , under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. [Copyright &y& Elsevier]