The Long Term Dynamic Behavior Of The Solution For Time-periodic Equation

The research content of this paper is divided into two parts.The first part,we study the stability of the zero solution for time-periodic linear perturbed systems.We can use the method of finding nonsingular and differentiable periodic matrices,by which the study on the properties of periodic linear systems can be reduced to the study on the properties of linear systems with constant coefficients.Then we extend the conclusion of the zero solution's stability of the constant coefficient linear system under small perturbation to the periodic linear system under small perturbations.Under certain conditions,if the perturbation term of a periodic linear disturbance system is of a higher order than ||x||,then the periodic linear disturbed system has the same stability as the zero solution of the undisturbed system.An illustrating example is provided at the end of this section.The second part,we use the viscosity solution theory proposed by Crandall et al in 80s,by which we propose to study the dynamics of the viscosity solution u(x,t)of the time 1-period Hamilton-Jacobi equation ut(x,t)+H(x,t,Dxu(x,t))=0.In this part,we use the ergodic approximation method to get the following conclusions:first,in the premise that H(x,t,p)is continuous and H(x,t,p)is coercive about p,by adding Lip condition for initial values,we prove there exist c ?c?R such that u(x,t)-ct is bounded from below and u(x,t)-ct is bounded from above on Tn×[0,?).Second,in the premise that H(x,t,p)is continuous and H(x,t,p)is coercive about p,by adding Lip condition of Hamilton function on time,we prove there exist c?R such that u(x,t)-ct is bounded from below on Tn×[0,?).