  It has been believed that spirals in pure stellar disks, especially the ones
spontaneously formed, decay in several galactic rotations due to the increase
of stellar velocity dispersions. Therefore, some cooling mechanism, for example
dissipational effects of the interstellar medium, was assumed to be necessary
to keep the spiral arms. Here we show that stellar disks can maintain spiral
features for several tens of rotations without the help of cooling, using a
series of high-resolution three-dimensional $N$-body simulations of pure
stellar disks. We found that if the number of particles is sufficiently large,
e.g., $3\times 10^6$, multi-arm spirals developed in an isolated disk can
survive for more than 10 Gyrs. We confirmed that there is a self-regulating
mechanism that maintains the amplitude of the spiral arms. Spiral arms increase
Toomre's $Q$ of the disk, and the heating rate correlates with the squared
amplitude of the spirals. Since the amplitude itself is limited by the value of
$Q$, this makes the dynamical heating less effective in the later phase of
evolution. A simple analytical argument suggests that the heating is caused by
gravitational scattering of stars by spiral arms, and that the self-regulating
mechanism in pure-stellar disks can effectively maintain spiral arms on a
cosmological timescale. In the case of a smaller number of particles, e.g.,
$3\times 10^5$, spiral arms grow faster in the beginning of the simulation
(while $Q$ is small) and they cause a rapid increase of $Q$. As a result, the
spiral arms become faint in several Gyrs.
