Summary: QUASILINEAR PARABOLIC PROBLEMS
VIA MAXIMAL REGULARITY
Herbert Amann
Institut f¨ur Mathematik, Universit¨at Z¨urich
Winterthurerstr. 190, CH­8057 Z¨urich, Switzerland
Abstract. We use maximal Lp regularity to study quasilinear parabolic
evolution equations. In contrast to all previous work we only assume
that the nonlinearities are defined on the space in which the solution
is sought for. It is shown that there exists a unique maximal solution
depending continuously on all data, and criteria for global existence are
given as well. These general results possess numerous applications, some
of which will be discussed in separate publications.
Introduction
In this paper we develop a general existence, uniqueness, continuity, and
differentiability theory for semilinear parabolic evolution equations of the
form
u + A(u)u = F(u) on (0, T), u(0) = x, (0.1)
where T is a given positive real number. This problem has already been
treated by many authors, including ourselves (e.g., [1], [13], [14], [16], and
the references therein). The main new feature of our present work, which dis-

Source: Amann, Herbert - Institut für Mathematik, Universität Zürich

Collections: Mathematics