Nonlinear Convective Instabilities of Fronts

Abstract:
co-author: Bjorn Sandstede
Through a case study, we investigate fronts that connect two different rest states. The rest state behind the front experiences a Turing instability: the essential spectrum of the linearization at the front crosses the imaginary axis. Using exponentially weighted norms, we see that, in the co-moving frame and on the linear level, the perturbations are pushed by the front to the left, away from the front. The problem with capturing this phenomenon on a nonlinear level is that the nonlinearity is not defined on exponentially weighted spaces. For our model equation we prove nonlinear stability of the front in exponentially weighted spaces, thereby establishing that the instability of the front is of a convective nature.