Asymptotic Regularity And Uniform Attractor Of A Class Of Non-autonomous Wave Equations

We consider the following initial and boundary value problem for a strongly non-autonomous wave-type evolutionary equation where 2≤r≤6,μ>0 andΩis an open bounded subsets of R3 with smooth boundary,u(x,t) is a unknown function, f(u) is the nonlinear function satisfies a critical exponential grownth condition and g is the given time-dependent external forcing term.There are two main difficulties of dynamic behavior for non-autonomous evo-lutionary equation.One is the Eq.(*) contains -Δutt,and the solution (uo,ut) depend on initial value haven't higher regularity, the dissipation and compactness of the system can't be obtained by energy inequality. One is the external force term g(t) depend on time is translation bounded and satisfy weak continuity. We will make use of extended Gronwall lemma and some proper analysis method or skill to prove the global dissipation and w-limit compact of the solution-semigroup of Eq.(*) where 2≤r≤6,then prove the existence of the uniform attractor.Therefore, in chapter 3,we have make use of extended Gronwall lemma and some proper analysis method or skill to prove the global dissipation andω-limit compact of the solution-semigroup of Eq.(*) where 2≤r≤6,then prove the existence of the globle attractor.In chapter 4, we obtain the existence and the structure of the uniform attractor through proving the asymptotic regularity of the solution where r=2,and the external force term is not translation-compact.