Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems
(2009)
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 are allowed to be nonlocal and to
have any growth. The main tools are new maximal regularity results [19, 20]
in Sobolev–Morrey spaces for linear parabolic initial boundary value problems
with nonsmooth data, linearization techniques and the Implicit Function Theorem.