随机偏微分方程作为随机分析的一个分支，广泛应用于物理学、力学、光学、数学、化学、通讯等许多领域，在人口统计、经济、金融等应用方面也发挥着重要作用.本文主要通过构造Lyapunov泛函，利用比较方法和Kaplan特征值法对两类随机偏微分方程的不变测度及爆破性进行研究.主要研究内容如下：

首先考虑了一类乘法噪声驱动下具有二阶记忆项的随机粘弹性波动方程.通过Lyapunov泛函技巧，获得了方程解的弱紧致性；证明了转移半群的bw-Feller性质，从而给出了不变测度的存在性理论，并给出该定理的一个实际应用的例子.

其次讨论了一类加法噪声驱动下的四阶随机波动方程.给出了相应四阶确定性方程解的爆破；通过比较方法，获得了四阶随机波动方程的解在非期望意义下爆破概率不为零，并给出了爆破时间的上界估计.

最后研究了双乘法噪声驱动下的非局部随机抛物方程.建立了局部弱解的存在唯一性；利用Kaplan特征值法，获得了局部弱解在期望意义下的爆破，并给出了爆破时间的上界估计.相比较单乘法噪声，双乘法噪声会加快爆破发生.

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.

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