On The Long-term Dynamical Behavior Of Weakly Damped Wave Equations

In this dissertation, we will consider the long-term behavior for the solution of following weakly damped wave equation in some proper phase space. where Ω is R3 or a bounded smooth domain of R3.This dissertation has two main parts. In the first part we study the exis-tence of global attractor for the solution of weakly damped wave equation with external source term(i.e. g ∈ H-1) for the case of bounded domain and R3 respec-tively. The first situation is critical nonlinearity. A regularity lifting argument is established thanks to the Strichartz inequality, then we can obtain the exis-tence and translational regularity of the global attractor. In the case of super critical nonlinearity, to obtain the uniqueness, we define a new class of solution called TR-solution. Due to the Strichartz inequality, the well-posedness can be obtained. In the sequel, in the case of Ω is bounded domain, we apply the energy method to verify the semigroup S(t) is asymptotically compact. And in the case of Ω=R3, after strengthening the dissipative condition appropriate, solutions are mainly concentrated in a big ball in R3, then according to Aubin-Lions lemma, we can also prove the asymptotic compactness. Hence the existence of global attractor is obtained.The second part is concerned with the long time behavior for the strong solution of weakly damped wave equation with sub-quintic nonlinearity. With the help of Strichartz estimate, we can establish an energy estimate for the strong solution. And then the global attractor and exponential attractor are obtained in (H2(Ω)∩H01(Ω))×H01(Ω).