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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 existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation . It proves that the Cauchy problem admits a global weak solution under the assumptions that , is of polynomial growth order, say p (>1), either , , where is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension . In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown. [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]

Abstract: The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when , where , and are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as , as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension , the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem. [Copyright &y& Elsevier]

Abstract: The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when , where , and are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension , the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if , and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown. [Copyright &y& Elsevier]