Front speeds in the vanishing diffusion limit for reaction-diffusion-convection equations

Travelling fronts for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity ε ≥ 0 exist for all velocities greater than or equal to an ε-dependent minimal value, both in the parabolic case when ε > 0 and in the hyperbolic case when ε = 0. We prove that as ε → 0, the minimal velocity c∗ε converges to c∗, the minimal value when ε = 0, and that c∗ε ≥ c∗ for all ε > 0. The proof uses comparison theorems and the variational characterization of the minimal parabolic front velocity. This convergence also yields a reaction-independent sufficient condition for the minimal velocity of the parabolic problem for small positive ε to be strictly greater than the value predicted by the problem linearized about the unstable equilibrium, that is, for the minimal-velocity travelling front of the viscous equation to be pushed for sufficiently small ε.