非线性双曲型随机偏微分方程及其相关研究

本项目将以物理、工程、材料力学及气动力弹性学中的一类非线性双曲型随机偏微分方程为主要研究对象。我们建立的模型不仅可应用于非线性随机波动方程，还可应用于周期边界条件、固定支承边界条件或铰直承边界条件下的非线性随机梁方程。我们将证明和解决由Lévy噪音驱动的双曲型随机偏微分方程的全局mild解的存在唯一性。并在此基础上进一步证明解的一致有界性、渐进稳定性和不变测度的存在性等问题。本项目也将研究在无穷维空间下随机卷积的极大值不等式。我们计划将随机卷积的极大值不等式推广到Banach空间下，从而可应用于Sobolev空间。我们的研究成果可以丰富随机偏微分方程、无穷维空间上随机分析的基础理论内容，同时我们在研究方法上的突破也将对不同类型方程之间的研究方法给予启示。

The present project aims at studying on nonlinear stochastic partial differential equations of hyperbolic type arising in physics, engineering, mechanics of materials and aeroelasticity. The results we obtained about this model are applicable to a wide class of equations including nonlinear stochastic wave equations and nonlinear stochastic beam equations subject to either periodic boundary conditions, or the clamped boundary conditions, or the hinged boundary conditions. We will prove the existence and uniqueness of global mild solutions to stochastic differential equations of hyperbolic type driven by Lévy noises. We will also prove the ultimate boundedness, asymptotic stability of the zero solution and the existence of invariant measure. Moreover, the maximal inequalities in infinite dimensional spaces will be investigated in this project. We will extend the maximal inequalities for stochastic convolutions to a more general Banach space setting, which will include Sobolev spaces. The completion of the proposed project will enrich the existing theoretical achievements on stochastic partial differential equations and stochastic analysis in infinite dimensional spaces, and the approach we have developed can provide a complementary perspective to the study of stochastic differential equations of various types.