The Research Of The Dynamics For Nonlinear Evolution Equations

This paper is devoted to studying the dynamic behavior and multiplicity of solutions for nonlinear evolution equations with Landesman-Laze type condition and the eventual stability of asymptotically autonomous systems with constraints.In the first chapter,we introduce the background and the main results of this paper.The second chapter is used to present some preliminaries used in this paper.In the third chapter,we consider the bifurcation of the heat equations near the resonance.First,we establish the invariant manifold for nonautonomous evolution equations under some smallness requirement of the Lipschitz constant.Based on this result,we investigate the nonautonomous heat equation by restricting it on the invariant manifold.Combing the techniques of the dynamic bifurcation,we derive the periodic solution bifurcation from infinity.In particular,we establish the attractor bifurcation from infinity for the reduced system of the autonomous heat equation.By analyzing the shape of the attractor,we prove there are at least two equilibrium solutions from infinity near any resonance for the heat equation.In the forth chapter,we establish the dynamic bifurcation for cooperative reaction-diffusion systems with Landesman-Lazer type condition via Conley-Morse theory.Based on this result,we obtain the multiplicity of equilibrium solutions.In the last chapter,we turn our attention to the asymptotically autonomous system with constraint.First,we try to establish a criterion on stability of e-quilibrium solutions for the limit system with constraint.Then we establish the eventual stability and instability for asymptotically autonomous system with con-straint near the equilibrium solutions of its limit system.As an example,we revisit an extreme ideology model proposed in the literature and give a more detailed description on the dynamics of the system.