Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity

In this paper, we prove various qualitative properties of pulsating traveling fronts in periodic media, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov type or general monostable nonlinearities. Besides monotonicity, the main part of the paper is devoted to the exponential behavior of the fronts when they approach their unstable limiting state. In the general monostable case, the logarithmic equivalent of the fronts is shown and for noncritical speeds, the decay rate is the same as in the KPP case. These results also generalize the known results in the homogeneous case or in the case when the equation is invariant by translation along the direction of propagation.