Perturbation theory in extended systems: Sharp nonlocal center manifolds and weakly quenched planar fronts

Abstract

We give novel perturbative results in two spatially extended contexts. First, we proveexistence of sharp center manifolds for spatially extended nonlocal systems with exponentially localized convolution kernels. Technically, the use of C0-based spaces allows for optimal regularity of the center manifold in a certain set of coordinates, as well as allowing for a simpler cutoff argument for existence of the manifold. As an application, we prove a sharp Lyapunov-Center theorem for a class of nonlocal systems. Second, we prove existence of fronts in a quenched planar Allen-Cahn equation, in the scaling regime where a weak inhomogeneity in the bistable region and the speed of the quenching line are of the same order. We find that the inhomogeneity and quenching speed induce a contact angle of the front, and describe its dependence on both parameters. Analysis relies on a farfield core decomposition made possible by Fredholm properties of the linearized operator at the stationary front in exponentially weighted spaces.