  In this letter, we study the limit behavior of the evolution of Tsallis
entropy in self-gravitating systems. The study is carried out under two
different situations, drawing the same conclusion. No matter in the energy
transfer process or in the mass transfer process inside the system, when
nonextensive parameter q is more than unity, the total entropy is bounded; on
the contrary, when this parameter is less than unity, the total entropy is
unbounded. There are proofs in both theory and observation that the q is always
more than unity. So the Tsallis entropy in self-gravitating system generally
exhibits a bounded property. This indicates the existence of global maximum of
Tsallis entropy. It is possible for self-gravitating systems to evolve to
thermodynamically stable states.
