Explosive Solutions of Stochastic Retarded Parabolic and Hyperbolic Differential Equations

随机缓滞抛物型和双曲微分方程的爆炸解

• 批准号: EP/I019987/1
• 负责人: Kai Liu
• 金额: $ 0.45万
• 依托单位: University of Liverpool
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI019987%2F1
• 关键词: Explosive; Solutions; Stochastic; Retarded; Parabolic

As is well-known, wave motion is one of the most commonly observed physical phenomena. The progression of water waves and the propagation characteristics of light and sound are familiar everyday experiences. As mathematical models, wave motions are usually described by partial differential equations of hyperbolic type. Solutions to nonlinear wave equations with certain polynomial nonlinearity tend to develop singularities in finite time. This means that these solutions exist only locally. It is therefore of interest to study the effects of random perturbation on the solution behavior of such equations. Regarding them as a stochastic equation of Ito type, the existence of a long-time solution of the equations was proved under a nearly linear growth condition on the state dependence in the noise term. For a stronger nonlinear noise term, it is plausible to anticipate an explosive solution. We can raise the following question: for a wave equation with a polynomial nonlinearity, how does a random perturbation affect the solution behavior? In general, there exists only a local solution. So it is practically important, e.g., the study of stability property, to find suitable conditions to ensure the existence of a global solution.Another important class of stochastic systems is stochastic partial differential equations of parabolic type. Some typical examples of this kind are, e.g., stochastic reaction-diffusion equations or stochastic Burgers equations. To account for the possibility of a noise induced explotion, we will try to find conditions on the initial state and nonlinear terms so as that there exist positive solutions whose mean square norm will blow up in finite time. It becomes apparent that the principle of causality is often only a first approximation to the true situation and that a more realistic model equation would include some of the past states of the system. On the other hand, the quantities we are interested in will not be predictable in advance but, rather, will exhibit an inherent variation that should be taken into account by the model. This is usually accomplished by allowing the model equation to be probabilistic in nature.In summary, in the proposed programme we shall study the explosive solutions of stochastic parabolic and hyperbolic differential equations with time delays. Due to its complexity, the problem will be attacked as the first step for some specific models such as stochastic wave or reaction-diffusion equations with time delays. We shall consider a class of stochastic retarded wave equations driven by spatially regular Wiener random field and stochastic retarded reaction-diffusion equations distinguishing both the bounded domain and whole space. To analyze these concrete equations, we will systematically employ the familiar tools in partial differential equations and stochastic analysis. Then we expect to show how these concrete results lead to the investigation of general nonlinear stochastic functional evolution equations in an infinite dimensional space setting.

众所周知，波动是最常观察到的物理现象之一。水波的传播过程和光声的传播特性是我们日常熟悉的经验。作为数学模型，波浪运动通常用双曲型偏微分方程来描述。具有一定多项式非线性的非线性波动方程的解在有限时间内趋向于发展奇点。这意味着这些解只存在于局部。因此，研究随机扰动对这类方程解的影响是很有意义的。将其视为伊藤型随机方程，在噪声项状态依赖的近似线性增长条件下，证明了方程长期解的存在性。对于较强的非线性噪声项，可以预测出爆炸解。我们可以提出以下问题：对于多项式非线性的波动方程，随机扰动如何影响解的行为？一般来说，只存在一个局部解。因此，寻找保证全局解存在的合适条件具有重要的实际意义，例如稳定性的研究。另一类重要的随机系统是抛物型的随机偏微分方程。这类方程的典型例子有随机反应扩散方程或随机Burgers方程。为了考虑噪声引起爆炸的可能性，我们将尝试寻找初始状态和非线性项的条件，使其均方范数在有限时间内爆炸。很明显，因果关系原理通常只是真实情况的初步近似，而更现实的模型方程应该包括系统过去的一些状态。另一方面，我们感兴趣的量是不能提前预测的，而是会表现出一种固有的变化，这种变化应该被模型考虑在内。这通常是通过允许模型方程本质上是概率性的来实现的。总之，在本课程中，我们将研究带时滞的随机抛物型和双曲型微分方程的爆炸解。由于其复杂性，对于一些特定的模型，如随机波动方程或具有时滞的反应扩散方程，将把该问题作为第一步来研究。考虑一类由空间正则Wiener随机场驱动的随机延迟波动方程和区分有界区域和整个空间的随机延迟反应扩散方程。为了分析这些具体的方程，我们将系统地使用偏微分方程和随机分析中熟悉的工具。然后，我们期望展示这些具体结果如何导致在无限维空间设置中的一般非线性随机泛函演化方程的研究。