The Existence Of Global Attractors For A Class Fourth Order Nonlinear Wave Equation

This paper studies the initial boundary value problems of a strongly damped wave equa-tions with nonlinear source term utt-△u+△2u+μut-α△ut-△utt= f(u),μ,α> 0, x∈Ω,t>0, u(?)t=0=u0, ut(?)t=0=u1, x∈Ω, u(?)aΩ= 0,whileΩis an open bounded domain with smooth boundary in Rn.By using variational methods, the thesis proves the global existence of weak solutions. And the asymptotic behaviour of the solution is obtained by using some important inequali-ties, Galerkin method and integral linear estimation. The results indicate that the solutions of the problem decay to zero with the time t approaches to infinity exponently. And by integral inequality, Sobolev embedding theorem and the equivalence theorem, this thesis proves the ex-istence of global attractors for the problem in the inner product space H01(Ω) x H01(Ω). Due to the equation contains many famous known mathematical physics model equations as special cases, this study has general meanings. What's more, the thesis studies a lot of dynamic be-haviors of the systems which are affected by many different factors. It provides possibilities to reveal much more behaviours of the more complex evolution systems. The problem in the thesis is more complex and common, but not without additional restrictions on the nonlinear terms or the initial data. Therefore, the better conclusion is given in this thesis comparing with the existing results.