论文标题
小界噪声对自主半线性微分方程双曲线的影响
The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations
论文作者
论文摘要
在这项工作中，我们研究了非自主随机/随机扰动下自主微分方程的双曲线的持续性。对于线性情况，我们研究非自主随机动力学系统的鲁棒性和指数二分法的存在。接下来，我们建立了非线性微分方程双曲线平衡持续性的结果。我们表明，对于每种自主的随机扰动，具有双曲平衡的自主性半线性问题，存在一个有界的\ textIt {随机双曲线解决方案}，对于相关的非线性非自治随机动力学系统。此外，我们表明这些随机双曲线溶液会融合到自主平衡。作为一种应用，我们考虑具有小非自主乘白噪声的半线性微分方程，例如，我们将抽象结果应用于强烈阻尼的波动方程。
In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded \textit{random hyperbolic solution} for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.
