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

Uniform bounds of families of analytic semigroups and Lyapunov linear stability of planar fronts