Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study

.We study the asymptotic stability of a front connecting two unstable states. Such a structure typically appears when the stable state behind a Fisher–Kolmogorov–Petrovskii–Piskunov front destabilizes when going through an essential Turing bifurcation, giving rise to oscillating patterns. Despite the instability of both end-states, we obtain for the first time stability of such a structure against suitably localized perturbations, with algebraic temporal decay \(t-3/2\). To deal with the instability behind the front, we simultaneously control the error in two different norms. In the first norm, enhanced diffusive decay is obtained at a linear level through pointwise resolvent estimates. In the second norm, better suited for nonlinear analysis, we show that the error stays bounded in time by use of mode filters.