Comparison and Stability of Solutions for a Neutron Transport Problem with Temperature Feedback

Abstract

A system of coupled equations arising from the neutron transport in a reactor system is investigated where the effect of temperature feedback is taken into consideration. Using the method of successive approximation and the notion of upper and lower solutions, two monotone sequences are constructed for the corresponding integral equations. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the system. This monotone convergence leads to an existence-comparison theorem in terms of the initial iteration as well as each of the succeeding iterations. Through suitable construction of the initial iteration the existence-comparison theorem is then used to investigate the stability and instability property of a steady-state solution. Sufficient conditions in terms of the physical parameters are given to ensure the stability and instability of the system, including some explicit stability and instability regions. It is also shown under suitable conditions on the same set of physical parameters that global solutions exist for one class of initial functions while they blow up in finite time for another class of initial functions. Characterizations of these two classes of initial functions are obtained. In some special feedback model, global solutions exist for all initial functions but they grow at a rate no less than the order of $\exp (\exp (\eta t))$ for some $\eta > 0$.