在无界区域上随机强衰减波动方程的整体吸引子

Global attractor for stochastic strongly damped semilinear wave equations on unbounded domain

研究了定义在无界区域上的具有非线性弱衰减项和可加噪声的强衰减波动方程的渐近动力行为.证明了与方程相关联的随机动力系统的整体吸引子的存在性.为此,首先证明了弱解及有界吸收集的存在性,然后利用适当的截断函数分解解的方法证明了渐近紧性.主要难点是由于区域的无界性,一些紧性结果不再有效.为克服此难点采用了方程解的分解方法.

In this paper we study the asymptotic dynamics for strongly damped wave equations with nonlinear weak damping and additive noise defined on unbounded domains.We prove the existence of the Global attractor for the random dynamical system associated with the equations.For this end,we first prove the existence of weak solutions and bounded absorbing sets,and then prove the asymptotic compactness by using the decomposition method of appropriate cut-off functions.The main difficulty of this paper is that,due to the unboundedness of the domain,some compactness results are not available.To overcome this difficulty we use the decomposition method of the solution to the equations.