Existence And Asymptotic Behavior Of Solutions For A Class Of Nonlinear Evolution Equations

This paper studies the long time behaviors of the following initial boundary value problemfor a class of high order nonlinear dispersive-dissipative wave equations where 2≤r≤6,μ> 0,andΩis an open bounded subset of 3 with smooth boundary ?Ω. u ( x , t )is the unknown function, and the nonlinear term f∈1( ) satisfies a critical exponential growth condition. g∈L2(Ω) is some given function.Equations (Q), which appears as a class of nonlinear evolution equations, like the strain solitary wave equation and dispersive-dissipative wave equation, is used to represent the propagation problems of lengthways-wave in non-linear Elastic rods and Ion-sonic of space transformation by weak nonlinear effect, and like Karman equation is used to represent the flow of condensability airs in the across velocity of sound district, and etc.Since equations (Q) contain terms ( ut r? 2ut )t andΔu tt, the dissipation and compactness of the system can not be obtained by energy inequality or secondary functional associated with standard Gronwall lemma. Therefore we will make use of extended Gronwall lemma and some proper analysis method or skill to solve this problem.Firstly, in chapter 3, the existence and uniqueness of the global weak solution of the equations (Q) are proved by Galerkin method and energy inequality, where the nonlinear term f satisfies a critical exponential growth condition.Secondly, in chapter 4, we have obtained the existence of the global attractors of the semigroup {S (t )}t≥0for the weak solution of equations (Q) in H 01 (Ω)×H01(Ω)by the asymptotical smoothness method.Finally, in chapter 5, we have proved the existence of the global attractors of the the method ofω-limit compactness.