Explosive Solutions of Stochastic Retarded Parabolic and Hyperbolic Differential Equations

众所周知，波动是最常见的物理现象之一。水波的运动以及光和声音的传播特性是我们熟悉的日常经验。作为数学模型，波动通常由双曲型偏微分方程描述。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于奇异。这意味着这些解决方案只存在于局部。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Ito型随机方程，在噪声项依赖于状态的近似线性增长条件下，证明了方程长时间解的存在性.对于更强的非线性噪声项，似乎可以预期爆炸性的解决方案。我们可以提出以下问题：对于一个具有多项式非线性项的波动方程，随机扰动如何影响解的行为？一般来说，只存在局部解。因此，它实际上很重要，例如，研究稳定性，寻找适当的条件，以确保整体解的存在。另一类重要的随机系统是随机抛物型偏微分方程。这种类型的一些典型例子是，例如，随机反应扩散方程或随机Burgers方程。为了解释噪声引起爆炸的可能性，我们将试图找到关于初始状态和非线性项的条件，以便存在正解，其均方范数将在有限时间内爆炸。很明显，因果关系原理往往只是真实情况的第一近似，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，我们感兴趣的量将无法提前预测，而是将表现出模型应该考虑的固有变化。这通常是通过允许模型方程是概率的natural.In总结，在拟议的计划，我们将研究随机抛物和双曲时滞微分方程的爆炸性解决方案。由于问题的复杂性，对于一些特殊的模型，如随机波动方程、时滞反应扩散方程等，问题往往被作为第一步来研究。考虑一类由空间正则Wiener随机场驱动的随机时滞波动方程和区分有界区域和全空间的随机时滞反应扩散方程。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，我们期望显示这些具体的结果如何导致一般的非线性随机泛函发展方程在无限维空间设置的调查。

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.