Nonexistence of Global Solutions for an Integro-differential System in Reactor Dynamics

Abstract

This paper is concerned with the instability behavior of an integrodifferential system arising in nuclear reactor dynamics. The spatial domain under consideration can be either bounded, subject to certain boundary conditions, or the whole space $R^N$ It is shown that if the physical parameter $\beta (x)$ is non-negative and $\beta (x) \ne 0$, which corresponds to positive feedback reactivity in the reactor system, then for certain classes of initial functions the corresponding solution of the initial boundary-value problem (or the Cauchy problem) grows unbounded in finite time. This blowing-up property holds for a large class of nonlinear functions, including the physically most interesting one, and for very small initial perturbations from its equilibrium solution. An explicit instability region as well as an upper bound for the finite escape time are obtained.