两类非线性随机波动方程的爆破性及渐近性研究

In this project, we study the explosion and the asymptotic behavior of solutions for two classes nonlinear stochastic wave equations. A class is the stochastic wave equation derived from physics, optics, and other fields to describe nonlinear wave propagation in random media. Another is the stochastic viscoelastic wave equation derived from physics or cybernetics to describe the matter particles position of elastic materials and viscous materials mixed model. When the damping is linear or without, we study the existence and uniqueness of attractors and invariant measure for stochastic viscoelastic wave equation driven by addition noise, and mainly consider the explosion of non-global solutions, the existence of global solutions, invariant measure, and the competition of the indices between source term with noise term for the two classes stochastic wave equations driven by multiplicative noise. When the damping is nonlinear, we give the existence and uniqueness of non-global solutions, and mainly consider the competition of various nonlinear terms to obtain the sufficient conditions of global existence or explosion for the two classes stochastic wave equations driven by multiplicative noise. Moreover, when the solutions are global, we also study the decay, the stability, and the existence and uniqueness of random attractors for the two classes wave equations. The study of this project will not only help to deepen understanding and awareness wave propagation in various random media, and reveal the effects of random media on the asymptotic properties of solution, but also it also has a very important theoretical significance.

本项目拟研究两类非线性随机波动方程解的爆破性及渐近性。一类来源于物理学、光学等领域中描述非线性波在随机介质中传播的随机波动方程，另一类是控制论中描述由弹性和粘性材料混合而成的模型中物质微粒位置的随机粘弹性波动方程。当阻尼项是线性阻尼或不含阻尼时，研究加法噪声驱动下粘弹性波动方程随机吸引子和不变测度的存在唯一性，重点研究乘法噪声驱动下这两类方程局部解的爆破性、全局解的存在性及不变测度等，讨论源项与噪声项之间的指数竞争关系；当阻尼为非线性时，目标给出乘法噪声驱动下这两类方程局部温和解的存在唯一性，重点研究各种非线性项的竞争关系，获得解全局存在或爆破的充分条件，当解全局存在时，进一步考虑这两类方程解的退化性、稳定性及随机吸引子的存在唯一性等。本项目的研究不仅有助于加深理解和认识自然界中非线性波在各种随机介质中传播的实质，揭示随机介质对解渐近性质的影响，而且问题本身的研究也具有非常重要的理论意义。

本项目主要研究了两类非线性随机波动方程解的爆破性及渐近性。一类来源于物理学、光学等领域中描述非线性波在随机介质中传播的随机波动方程，另一类是控制论中描述由弹性和粘性材料混合而成的模型中物质微粒位置的随机粘弹性波动方程。首先利用迭代和停时技巧获得了乘法噪声驱动下带有非线性阻尼的随机粘弹性波动方程温和解的存在唯一性，利用Lyapunov泛函给出解全局存在的条件，利用凸函数方法和能量不等式给出充分条件使得：要么解在 L^2模意义下以正的概率有限时刻爆破，要么均方解有限时刻趋向无穷；给出阻尼项与源项之间非线性指数的竞争关系，并获得噪声项对爆破起阻止作用；其次，我们研究了一类非高斯Lèvy驱动下随机粘弹性波动方程不变测度的存在性；同时，利用黎曼几何方法我们也给出了变系数的粘弹性波动方程解的稳定性；研究了有界区域上三维的Navier-Stokes方程回拉吸引子的存在性；研究了非局部椭圆方程解的存在唯一性。本项目的研究不仅有助于加深理解和认识自然界中非线性波在各种随机介质中传播的实质，揭示随机介质对解渐近性质的影响，而且问题本身的研究也具有非常重要的理论意义。