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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 behavior of the Kirchhoff type equation with strong damping on . It shows that the related continuous semigroup possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. [Copyright &y& Elsevier]