Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

Reaction–diffusion equations on the real line that contain a control parameter are investigated. Of interest are travelling front solutions for which the rest state behind the front undergoes a supercritical Turing or Hopf bifurcation as the parameter is increased. This causes the essential spectrum to cross into the right half plane, leading to a linear convective instability in which the emerging pattern is pushed away from the front as it propagates. It is shown, however, that the wave remains nonlinearly stable in an appropriate sense. More precisely, using the fact that the instability is supercritical, it is shown that the amplitude of any pattern that emerges behind the wave saturates at some small parameter-dependent level and that the pattern is pushed away from the front interface. As a result, when considered in an appropriate exponentially weighted space, the travelling front remains stable, with an exponential in time rate of convergence.