Short-time existence theory toward stability for nonlinear parabolic systems

Abstract

We establish existence of classical solutions for nonlinear parabolic systems in divergence form on ${\mathbb{R}^n}$, under mild regularity assumptions on coefficients in the problem, and under the assumption of Hölder continuous initial conditions. Our analysis is motivated by the study of stability for stationary and traveling wave solutions arising in such systems. In this setting, large time bounds obtained by pointwise semigroup techniques are often coupled with appropriate short time bounds in order to close an iteration based on Duhamel-type integral equations, and our analysis gives precisely the required short time bounds. This development both clarifies previous applications of this idea (by Zumbrun and Howard) and establishes a general result that covers many additional cases.