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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 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 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]

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]

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 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]

In this paper, the global existence of solutions to the initial boundary value problem for a class of quasi-linear wave equations with viscous damping and source terms is studied by using a combination of Galerkin approximations, compactness, and monotonicity methods. [Copyright &y& Elsevier]

The paper studies the blowup of solutions to the initial boundary value problem for the “bad” Boussinesq-type equation utt−uxx−buxxxx=σ(u)xx, where b>0 is a real number and σ(s) is a given nonlinear function. By virtue of the energy method and the Fourier transform method, respectively, it proves that under certain assumptions on σ(s) and initial data, the generalized solutions of the above-mentioned problem blow up in finite time. And a few examples are shown, especially for the “bad” Boussinesq equation, two examples of blowup of solutions are obtained numerically. [Copyright &y& Elsevier]

The paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved Boussinesq type equation utt−uxx−uxxtt=σ(u)xx. By a Galerkin approximation scheme combined with the continuation of solutions step by step and the Fourier transform method, it proves that under rather mild conditions on initial data, the above-mentioned problem admits a unique generalized solution u∈W2,∞([0,T];H2(0,1)) as long as σ∈C2(R). In particular, when σ(s)=asp, where a≠0 is a real number and p>1 is an integer, specially a<0 if p is an odd number, the solution blows up in finite time. Moreover, two examples of blowup are obtained numerically. [Copyright &y& Elsevier]

The paper investigates the well-posedness, regularity and longtime behavior of solutions for an extensible beam equation with fractional rotational inertia and structural nonlinear damping: (1 + (− Δ) θ) u t t + Δ 2 u − M (‖ ∇ u ‖ 2) Δ u + N (‖ ∇ u ‖ 2) (− Δ) α u t + f (u) = g , with dissipative index α ∈ (0 , 1 ] and rotational index θ ∈ [ 0 , α). We find a new critical exponent p θ α ≡ N + 2 (2 α − θ) (N − 4) + (> p ⁎ = N (N − 4) + ) depending on both θ and α and show that when the growth exponent p of the nonlinearity f (u) is up to the range: 1 ≤ p < p θ α , the well-posedness, regularity and longtime behavior of the solutions are like parabolic, in the concrete, (i) the IBVP of the equation admits a unique strong solution while its initial data is in weaker energy space; (ii) the related solution semigroup has a strong global attractor and a strong exponential attractor for each θ ∈ [ 0 , α) , where the "strong" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the strong solution space; (iii) the family of strong global attractors { A θ } θ ∈ [ 0 , α) is upper-semicontinuous on the perturbed index θ at any point θ 0 ∈ [ 0 , α). These results improve and extend those in recent literatures [17,24]. [ABSTRACT FROM AUTHOR]