The Longtime Dynamics Of The Nonlinear Wave Equations With Weakly Damping

In this thesis,we mainly research the global well-posedness and the longtime dynamic behavior of the following nonlinear weakly dissipative wave equations:Firstly,we investigate the initial value problem of the weakly damped au-tonomous wave equation(1)with a sup-cubic nonlinearity in locally uniform s-paces.We prove the global existence and uniqueness of Shatah-Struwe solutions by utilizing the recent progress in Strichartz estimates for the linear wave equa-tion in bounded domains.Due to the unboundedness of domain,the supercritical nonlinearity and the inherent characteristic for the equation,we develop further the so-called contractive function method introduced in[32,48,73]and deduce the asymptotic compactness of Eq.(1).Further,we establish the existence of(Hlu1(R3)×Llu2(R3),Hρ1(R3)×Lρ2(R3))-global attractor for the dynamical system of Eq.(1).In addition,we established the Strichartz type estimates for the solu-tions of Eq.(1)in locally uniform spaces.This enriches and develops the theory of Strichartz type estimates for wave equations.Secondly,in the non-autonomous case,we study the global well-posedness and the longtime dynamic behavior of Eq.(1)with the supercritical nonlinearity and the non-translation compact external force in a bounded domain,mainly in-cludes:1)By using the translation boundedness of the external force,as well as the Strichartz estimates for the case of bounded domains to prove that the global well-posedness of Shatah-Struwe solutions for the quintic weakly damped non-autonomous wave equation in natural energy space.2)We prove the processes of Shatah-Struwe solutions are weakly continuous in a certain topology.3)We estab-lish the existence and structural characterization of a strong uniform attractor for Eq.(1).Considering the effects of the sup-cubic nonlinearity,the non-translation compact external force and the inherent characteristic for wave equations,com-bined with the convergence result of the class of time regular external forces(see Theorem 4.3.1),we prove the uniformly asymptotic compactness of the dynamical systems by applying the contractive function method for the non-autonomous hy-perbolic type evolution equations.Simultaneouslyl,we give the other proof of the uniformly asymptotic compactness of the dynamical systems when the nonlineari-ty is super-cubic by utilizing the so-called energy method that generalized by S.V.Zelik[85]for dealing with the case of non-translation compact external forces.Finally,we interested in the dynamical behavior of the sup-cubic weakly damped non-autonomous wave equation(1)in locally uniform spaces.When the external force g∈Lb2(R;Llu2(R3))depends on time t explicitly,we establish the global well-posedness of Shatah-Struwe solutions for Eq.(1).Since the unbound-edness of domain and the supercritical nonlinearity,the compactness will lost in the general sense.To this end,we construct the energy inequalities for the Shatah-Struwe solutions,and then,use the contractive function method to establish the(Hlu1(R3)×Llu2(R3),Hρ1(R3)×Lρ2(R3))-pullback asymptotic compactness of the pro-cesses.Further,we prove the existence of a(Hlu1(R3)×Llu2(R3),Hρ1(R3)×Lρ2(R3))-pullback attractor based on the criterion for the existence of the attractors for the non-autonomous systems.