  Many monostable reaction-diffusion equations admit one-dimensional travelling
waves if and only if the wave speed is sufficiently high. The values of these
minimum wave speeds are not known exactly, except in a few simple cases. We
present methods for finding upper and lower bounds on minimum wave speed. They
rely on constructing trapping boundaries for dynamical systems whose
heteroclinic connections correspond to the travelling waves. Simple versions of
this approach can be carried out analytically but often give overly
conservative bounds on minimum wave speed. When the reaction-diffusion
equations being studied have polynomial nonlinearities, our approach can be
implemented computationally using polynomial optimization. For scalar
reaction-diffusion equations, we present a general method and then apply it to
examples from the literature where minimum wave speeds were unknown. The
extension of our approach to multi-component reaction-diffusion systems is then
illustrated using a cubic autocatalysis model from the literature. In all three
examples and with many different parameter values, polynomial optimization
computations give upper and lower bounds that are within 0.1% of each other and
thus nearly sharp. Upper bounds are derived analytically as well for the scalar
RD equations.
