Nonlinear stability of fast invading fronts in a Ginzburg-Landau equation with an additional conservation law
In this talk, I consider the stability of traveling fronts connecting an invading state to an unstable ground state in a Ginzburg-Landau equation with an additional conservation law. This system appears generically as an amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the Bénard-Marangoni convection problem. The main result is the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. After presenting the general strategy, I discuss the specific challenges faced in the proof, namely the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.