Explosive Solutions of Stochastic Retarded Parabolic and Hyperbolic Differential Equations

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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.