The Existence Of Uniform Attractor For A Class Of Non-autonomous Wave Equation

In this thesis,we consider the long-time behavior of the solutions for the following evolutionary equation: whereμ≥0 is a constant,Rr= (τ,+∞),τ∈R,Ω(?)R3 is abounded domain with smooth boundary,u(x, t) is a unknown function, and ut=(?)h(·) is known function, f(·) is the nonlinearity and satisfies some suitable conditions,g is a time-dependent forcing.To test and verify the existence of for the non-autonomous dynamic systems, we have to overcome two main difficulties. One is to obtain the dissipation of systems, that is to test and verify the existence of a bounded uniformly absorbing set for sys-tems; another is to obtain some compact of system, that is to test and verify system with uniformly asymptotically compact or the uniformω-limit compact.Because we mainly consider the long-time behavior for the strongly solutions of the systems (0.1), it is difficult to obtain the higher regularity of the solutions (u, ut) relative to initial values；otherwise,the time-dependent forcing g is the classes of functions denoted by Lc*2(R; L2(Ω)) which are translation non-compact, it is very difficult to obtain the com-pactness of equations(0.1).In the paper,we use the promotion of Growall inequality to overcome the first problem, so we obtain the bounded absorbing sets of the sys-tem (0.1);to get the uniform w-limit compactness of the non-autonomous systems,we make use of the analytical methods which are similar to the energy estimation.In this paper,we only assume that the nonlinear term satisfies the general con-ditions.In the third chapter,we investigate the existence of uniform attractor for sys-tem(0.1) in(?)1,here h(ut,Δut)=-Δut>,μ>0. In the forth chapter,we investigate the existence of uniform attractor for system(0.1) inεhere h(ut,Δut)=-λΔut,μ=0 .In the fifth chapter,we investigate the existence of global attractor for system(0.1) inH1(R3)×H1(R3),here h(βut-Δut,μ> 0,g(x, t)=g(x).