Invariant Measure And Blow-up Of Solutions Of Stochastic Partial Differential Equations

As a branch of stochastic analysis,stochastic partial differential equation is widely used in many fields,such as physics,mechanics,optics,mathematics,chemistry,communication and so on.It also plays an important role in demographic,economic,financial and other applications.This paper mainly studies the invariant measure and blow up of two kinds of stochastic partial differential equations by constructing Lyapunov functional method,comparison method and Kaplan eigenvalue method.The main research contents are as follows:First,a class of stochastic viscoelastic wave equations with damping driven by multiplicative noise are considered.By Lyapunov functional technique,the weak compactness of the solution of the equation is obtained;The bw-Feller property of the transfer semigroup is proved,the existence theory of invariant measure is given.Finally,and an example of practical application of the theorem is given.Second,a class of fourth-order stochastic wave equations driven by additive noise is discussed.The blow up results of the solutions for the corresponding fourth-order deterministic equations are given.By comparison method,the blow up probability which are not zero of the solutions for the fourth-order stochastic wave equations is obtained in the unexpected sense,and the upper bound estimation of the blow up time is given.Third,a class of nonlocal stochastic parabolic equations driven by double multiplicative noise is studied.The existence and uniqueness of local weak solutions are established;Using Kaplan eigenvalue method,blow up of the local weak solutions are obtained in the expected sense,and the upper bound of blow up time is given.Compared with single multiplicative noise,double multiplicative noise can accelerate the occurrence of blowing up.