  We study families of analytic semigroups, acting in a Banach space, and
depending on a parameter, and give sufficient conditions for existence of
uniform with respect to the parameter norm bounds using spectral properties of
the respective semigroup generators. In particular, we use estimates of the
resolvent operators of the generators along vertical segments to estimate the
growth/decay rate of the norm for the family of analytic semigroups. These
results are applied to prove the Lyapunov linear stability of planar traveling
waves of systems of reaction-diffusion equations, and the bidomain equation,
important in electrophysiology.
