  We propose a physically sensible formulation of initial value problem for
black hole perturbations in higher-order scalar-tensor theories. As a first
application, we study monopole perturbations around stealth Schwarzschild
solutions in a shift- and reflection-symmetric subclass of degenerate
higher-order scalar-tensor (DHOST) theories. In particular, we investigate the
time evolution of the monopole perturbations by solving a two-dimensional wave
equation and analyze the Vishveshwara's classical scattering experiment, i.e.,
the time evolution of a Gaussian wave packet. As a result, we confirm that
stealth Schwarzschild solutions in the DHOST theory are dynamically stable
against the monopole perturbations with the wavelength comparable or shorter
than the size of the black hole horizon. We also find that the damped
oscillations at the late time do not show up unlike the ringdown phase in the
standard case of general relativity. Moreover, we investigate the
characteristic curves of the monopole perturbations as well as a static
spherically symmetric solution with monopole hair.
