Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain Lp-behavior (p⩾1) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in L1∩H2 recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations |u0|⩽E0e−|x|2/M, |u0|H2⩽E0 and |u0|⩽E0(1+|x|)−r, r>2, |u0|H2⩽E0 respectively, E0>0 sufficiently small and M>1 sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.