The Study Of Long Time Behavor Of Wave Equation

In this doctoral thesis, we study the long time behavior for the following two classes of dissipative wave equations with initial boundary conditions we obtain the existence of time-dependent global attractor for the first equations and the new properties of the global attractor for the second equations, where ε(t)â†'0, as tâ†'oo in the first equations.I. Existence of attractors.For the first equations,we introduce the following energy functional naturallywhich exhibits a structural dependence on time. Furthermore, it is easy to realize that the vanishing character of ε at infinity alters uniform boundedness and asymptotic compactness of the system in the phase space X=H01(Q)×L2(Q) in the usual sense. Hence, we introduce a time-dependent family of spaces H01(Ω)×Lt2(Ω), and establish the process{U(t,Ï„)}t>Ï„ corresponding to the solutions for the first problem in the s-paces. In order to obtain the time-dependent global attractor, we present a concept of asymptotically compact for the process in time-dependent space and establish a crite-rion for the existence of time-dependent global attractor. To overcome the difficulties brought about by the critical exponent of the nonlinearity, we establish a technical method for verifying asymptotic compactness via contractive function. Finally, we ap-ply these results to the first equation and obtain the existence of the time-dependent attractors for the equations.â…¡:Property of attractors.For the second equations, according to the latest theory about the property of the global attractor for symmetric dynamics ([139]), we could estimate the lower bound of Z2index of the global attractor and get the existence of the multiple equilibrium points in the global attractor for the symmetric dynamical systems, therefore, we could reveal the property of the global attractor in some ways.Due to the origin is the minimal point of the corresponding Lyapunov functional of problem (â…¡) when λ=0, while λ is large enough, the origin is no longer the mini-mal point, we will discuss it from two aspects, first, the orgin as the equilibrium of the semigroup is stable, i.e. the origin is the local minimal point of the Lyapunov function for the equation, we establish some criterions to verify the small neighborhood of the origin is an attracting domain, and then obtain the existence of multiple equilibrium point in the global attractor for the second equations and give the estimate of the di-mension for the attractor according to the latest theory from [139]; second, the orgin as the equilibrium of the semigroup is unstable, hence the origin is not the local minimal point of the Lyapunov function for the equation, by expanding the intersection lem-ma, and then establish the abstract theorem, finally by applying the abstract theorem we obtain the existence of multiply equilibrium points in the global attractor for the second equations and give the estimate of the dimension for the attractor.