Continuity And Differentiability Of Semigroups Of Solutions For Nonlinear Evolution Equations

The existence of attractors for Semigroups of solutions of Nonlinear Evolution Equations is a hot research field in recent years.The continuity and differentiability of Semigroups of solutions to Nonlinear Evolution Equations are important links in the study of the existence of attractors and exponential attractors.In this paper,we mainly discuss the continuity and differentiability of Semigroups of solutions for Nonlinear Evolution Equations in different dimensional spaces.It is divided into three chapters.The first chapter studies the follow three equations in a bounded domain of R,If(0.01)satisfies f’(s)<l,f∈C2(Ω),g∈C∞(Ω)we can deduce its solution semigroup is continuous in L2(Ω).The images of solutions of(0.0.2)and(0.0.3)are made to prove continuous dependence of the solutions on the initial values.The second chapter studies the follow four equations in a bounded domain of R2 We derive that if the nonhomogeneous terms of(0.0.4)(0.0.5)(0.0.6)(0.0.7)satisfy certain conditions respectively,then the solution semigroup of(0.0.4)is differentiable in Lq(Ω),H01(Ω),H2(Ω).The solution semigroup of(0.0.5)is continuous in H01(Ω)×L2(Ω).The solution semigroup of(0.0.6)is continuous in H01(Ω).The solution semigroup of(0.0.7)is continuous in(L2(Ω)× L2(Ω),H01(Ω)× H01(Ω)).Under the hypothesis of Ω is a bouuded domain in R3,In Chapter 3,we discuss the continuity of the solution Semigroups of swift Hohenberg equations of(0.0.4)and(0.0.5)in the corresponding space,the continuity of solution semigroup of(0.0.6)in L2(Ω)and H01(Ω),the differentiability of solution semigroup of the Weakly damped form of(0.0.5)inH01 × L2(Ω).