  We study the heating of the cool cores in galaxy clusters by cosmic-rays
(CRs) accelerated by the central active galactic nuclei (AGNs). We especially
focus on the stability of the heating. The CRs stream with Alfv\'en waves in
the intracluster medium (ICM) and heat the ICM. First, assuming that the
heating and radiative cooling is balanced, we search steady state solutions for
the ICM and CR profiles of clusters by solving a boundary value problem. The
boundary conditions are set so that the solutions are consistent with
observations of clusters. We find steady state solutions if the magnetic fields
are strong enough and the association between the magnetic fields and the ICM
is relatively weak. Then, we analyse the stability of the solutions via a
Lagrangian perturbation analysis and find that the solutions are globally
stable. We confirm the results by numerical simulations. Using the steady state
solutions as the initial conditions, we follow the evolution of the profiles
for 100 Gyr. We find that the profiles do not evolve on time scales much larger
than cluster lifetimes. These results, as well as consistency with observations
of radio mini-halos, suggest that the CR heating is a promising mechanism to
solve the so-called "cooling flow problem".
