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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]

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]