The Long-time Behavior Of Solutions For Two Types Of Nonlinear Evolutionary Equations On Non-cylindrical Domains

With the development of society and technology,there are more and more partial differential equations on non-cylindrical domains coming from physics,bi-ological mathematics and control theory,etc.Due to their inherent characteristics and the lag of development for non-autonomous dynamical systems,there are a few research on the dynamics of evolution equations on non-cylindrical domains.In this paper,we considered the dynamics of the complex Ginzburg-Landau(CGL)equations and the nonlinear weakly dissipative wave equations on two kinds of concrete non-cylindrical domains.Firstly we studied the dynamics for the CGL equations and nonlinear weakly dissipative wave equations on non-cylindrical domains that the spatial domains Ot are obtained from a bounded base domain O by a diffeomorphism respectively,mainly includes:(1)Using a suitable change of variables and some established important inequalities,we established the existence and uniqueness of strong so-lutions for these two types of equations.(2)We gave a definition of the weak solutions by selecting a proper function space,and proved the results of the exis-tence and uniqueness of weak solutions.(3)Due to the domain is varying and the inherent characteristics for the two types of the equations,we applied finite ?-net and contraction function methods to obtain the asymptotic compactness of the two dissipative systems respectively.These are the main contents of chapter 3 and chapter 5.At the same time,we introduced some new methods and techniques to establish the existence of pullback attractor for the dynamical systems,which can improve the restriction on varying domains in the aforementioned works on the topic,such as[67,102].In addition,We also constructed a sufficient condition to ensure the transformation is hyperbolic in Appendix of chapter 5,which may be of independent interest.Secondly we investigated the dynamics of the CGL equations on non-cylindrical domains that the spatial domains Ot satisfying a monotonicity condition.It is well known that the classical method to study this type region is the "penalty method".However,the existing penalty function cannot be applied directly to the study of CGL equations due to their inherent characteristics.Combining with the existing penalty function and the characteristics of CGL equations,we presented a new type of penalty function.In particular,this method will play an important role in studying the dynamics of other dissipative equations on such non-cylindrical domains,such as wave equations.At the same time,we proved the uniqueness of weak solutions using the choosing method initiated by S.Bonaccorsi[14],and then we established the existence of a absorbing set in H01(Ot)which lead to the existence of a D-pullback attractor in L2(OT)directly.Finally we listed some problems which will be considered in forthcoming ar-ticles based on the results and the current situation.This research work,no matter for the source of the problems,or for the devel-opment of the theory of infinite dimensional dynamical systems and applications,will play a positive role in promoting for the dynamics.