Long-time Dynamical Behavior For Two Classes Of Dissipative Systems With Degenerate Operator

In this doctoral dissertation,we consider the long-time dynamical behavior of two classes of dissipative degenerate equations:the degenerate parabolic equation and the degenerate hyperbolic equation.By using the attractor theory of the infinite dimensional dynamical system and combining the characteristics and difficulties of the equation,we establish some new a priori estimates of the equation and obtain a series of novel results.For the parabolic equations with degenerate operator ?_?,we consider the long-term behavior of the solution of the equation when the nonlinear term of the equation satisfies the critical growth.The difficulty of the problem lies in the nature of the degenerate operator ?_? and the lack of compactness caused by the critical nonlinear term.We decompose the nonlinear term and obtain the a priori estimates of the solutions of the corresponding equations by using the theory of compacness.Therefore,we obtain the global existence of attractor in W_?^1,2???for the semigroup generated by the solution of the parabolic equation by using the theory of attractors.For the degenerate hyperbolic equation^,we consider the hyperbolic equation with the constant damping coefficient and the hyperbolic equation with the time-dependent damping coefficient.Since the hyperbolic equation has weak dissipative properties and the solution operator lacks the regularization effect,so the hyperbolic equation has inherent difficulties compared with the parabolic equation.In the fourth chapter,we study the long-term behavior of the solution of the damped wave equation with the degenerate operator under the critical nonlinear term.After using the theory of operator semigroups we obtain the existence and uniqueness of the weak solution of the equation,and in order to overcome the weak dissipative property of the wave equation,we use the gate function method to obtain the absorbing set of the weak solution of the equation.Finally,we use the operator decomposition method to overcome difficulty caused by the critical nonlinear term and prove the asymptotic compactness.Further we prove that there exists a global attractor in X^1/2×L²???for the semigroup generated by the solution of the hyperbolic equation.In Chapter 5,we study the long-term behavior of solutions of hyperbolic equations with time-dependent damping coefficient under critical nonlinear term.The difficulty of the problem lies mainly in the nature of the space itself caused by the degenerate operator X-elliptic operator7,the dissipation property of the equation caused by the negative damping coefficient and the the lack of compactness caused by the critical nonlinear term.In particular,the fact that the damping coefficient may be negative makes it difficult to prove the dissipation in a general way.Therefore,in this dissertation we propose some additional technical assumptions about the damping coefficient.As far as we know,this is the first time to study the weakly damped wave equation with time-dependent and partially negative damping coefficient by using the theory of infinite dynamical systems.For this difficulty caused by partially negative damping coefficient,we obtain the bounded absorbing set by estimating the energy norm of the solution over parts of timeline.Finally,we decompose the equation into two parts and obtain that one part is dissipative and the other part is smoothing which show that the solution of the equation has asymptotic compactness.So we prove that there is a pullback attractor in X^1/2×L²???of the process generated by the solutions of the hyperbolic equation,and furthermore we prove the finiteness of the fractal dimension of the pullback attractor by the properties obtained over the decomposed equations.