Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.