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。水波的行进，光和声的传播特性，都是我们熟悉的日常经验。波动作为数学模型，通常用双曲型偏微分方程组来描述。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于奇性。这意味着这些解决方案仅存在于本地。因此，研究随机扰动对这类方程的解行为的影响是很有意义的。将其视为一个Ito型随机方程，在噪声项中状态依赖的近线性增长条件下，证明了该方程长时间解的存在性。对于更强的非线性噪声项，可能会出现爆炸性解。我们可以提出以下问题：对于具有多项式非线性的波动方程，随机扰动是如何影响解的行为的？一般来说，只存在一个局部解决方案。另一类重要的随机系统是抛物型随机偏微分方程解的存在。这类问题的一些典型例子是随机反应扩散方程或随机Burgers方程。为了考虑噪声诱导爆炸的可能性，我们将试图找到关于初态和非线性项的条件，以便存在均方范数在有限时间内爆破的正解。很明显，因果关系原理往往只是对真实情况的第一近似值，一个更现实的模型方程将包括系统的一些过去状态。另一方面，我们感兴趣的数量将不是预先可预测的，而是会显示出模型应该考虑的内在变化。这通常是通过允许模型方程的概率性质来实现的。综上所述，在拟议的程序中，我们将研究随机抛物型和双曲型时滞微分方程的爆炸解。由于它的复杂性，对于某些特定的模型，如随机波或具有时滞的反应扩散方程，该问题会被作为第一步来攻击。我们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域和整个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组和随机分析中熟悉的工具。然后，我们期望展示这些具体的结果如何导致无限维空间中一般的非线性随机泛函发展方程的研究。

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.