Summary: NONLOCAL QUASILINEAR PARABOLIC EQUATIONS
HERBERT AMANN
Abstract. We give a survey of the most common approaches to quasilinear
parabolic evolution equations, discuss their advantages and drawbacks, and
present an entirely new approach based on maximal Lp regularity. Our gen­
eral results apply, above all, to parabolic initial boundary value problems being
nonlocal in time. This is illustrated by indicating their relevance for quasilin­
ear parabolic equations with memory and, in particular, for time regularized
versions of the Perona­Malik equation of image processing.
Dedicated to S.M. Nikols'kii on the occasion of his 100 th birthday
Introduction
In this paper we discuss a new approach to the abstract quasilinear parabolic
equation
—
u +A(u)u = F (u) on (0, T ), u(0) = u 0 , (0.1)
where T is a fixed positive number. Formulation (0.1) encompasses a great variety
of concrete problems, most prominently parabolic initial boundary value problems
of the form
# t u -# · # a(u)#u # = f(u)
on# × (0, T ),

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

Collections: Mathematics