Topological Structure Of Solution Sets And Long-time Behavior For Several Types Of PDEs With Multi-valued Right-hand Sides

Multi-valued partial differential equations are also called partial differential inclusions or evolution equations with multi-valued right-hand sides,which are characterized by multi-valued operators in the principal part or multi-valued nonlinearity.They are used in various fields of biological,physical and engineering applications such as control systems where control factors are taken in the form of feedbacks.A partial differential equation with multi-valued perturbation is a notion which generalizes an ordinary differential equation.Therefore,all problems considered for differential equations such as existence of solutions,continuity of solutions,dependence on initial conditions and parameters,are present in the theory of multi-valued partial differential equations.Since a partial differential equation with multi-valued perturbation usually has many solutions starting at a given point,new research contents appear,such as topological structure of solution sets and long time behaviors of multi-valued dynamical systems.In this paper,we study the topological structure of solution sets for two kinds of multi-valued partial differential equations with impulsive effects and the global attractor for a spe-cial system with multi-valued perturbation.Our results will be divided into three parts.Part 1.We study the topological structure of solution set for a third-order(in space)partial differential equation with multi-valued perturbation and impulsive effects.In the framework of the infinite-dimensional space,the differential operator in the principal part corresponds to an Airy operator generating a noncompact C₀-group of unitary operators.Our attention is paid to characterizing R_?-structure of the solution set.Geometric aspects of the corresponding solution map are also considered.In our main results,no compactness condition on the impulsive functions is needed.Moreover,we give illustrating examples of the nonlinearity and impulsive functions.Part 2.We are devoted to study the topological structure of solution set for a Volterra-type nonautonomous multi-valued partial differential equation with impulsive effects.The evolution family generated by the operators in the principal part is of no equicontinuity.Our attention is paid to establishing the R_?-structure of the solution set and geometric features of the corresponding solution map.Moreover,the long-time existence of the corresponding nonlocal Cauchy problem is also considered.Finally,we present an example to illustrate the applicability of our abstract results.Part 3.We consider a class of partly dissipative reaction-diffusion system with discon-tinuous nonlinearity.The main characteristic of the system is that the linear part is not the subdifferential of a compact-type function.We first present an existence result on global solutions.Then,we prove that the system possesses a global attractor having the H^r×H^r-regularity(0?r<2).Moreover,by showing the Kneser property for the system,the global attractor is proved to be connected.