  We study the axial quasinormal modes of hairy black holes in Gauss-Bonnet
gravity with massive self-interacting scalar field. Two coupling functions of
the scalar field to the Gauss-Bonnet invariant are adopted with one of them
leading to black hole scalarization. The axial perturbations are studied via
time evolution of the perturbation equation, and the effect of the scalar field
mass and the self-interaction constant on the oscillation frequency and damping
time is examined. We study as well the effect of nonzero scalar field potential
on the critical point at which the perturbation equation loses hyperbolicity in
the case of black hole scalarization. The results show that the non-zero scalar
field potential extends the range of parameters where such loss of
hyperbolicity is observed thus shrinking the region of stable black hole
existence. This will have an important effect on the nonlinear dynamical
simulation studies in massive scalar Gauss-Bonnet gravity.
