  In this paper, we consider the large-time behavior of solutions of a reaction
diffusion system arising from a nuclear reactor model with the Robin boundary
conditions, which consists of two real-valued unknown functions. In particular,
we show that global solutions of this system are uniformly bounded in a
suitable norm with respect to time. Since this system has no variational
structure, we cannot apply the standard methods relying on the Lyapunov
functional in order to obtain a priori estimates of global solutions. To cope
with this difficulty, we make use of the weighted $L^1$ norm characterized by
the first eigenfunction of Laplacian with the Robin boundary condition.
