Linear and nonlinear marginal stability for fronts of hyperbolic reaction diffusion equations

Abstract

We study travelling fronts of equations of the form $u_{tt} + ϕ(u) u_x = u_{xx} + f(u)$. A criterion for the transition from linear to nonlinear marginal stability is established for positive functions $ϕ(u)$ and for any reaction term $f(u)$ for which the usual parabolic reaction diffusion equation $u_t = u_{xx} + f(u)$ admits a front. As an application, we treat reaction diffusion systems with transport memory.