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NONLOCAL QUASILINEAR PARABOLIC EQUATIONS HERBERT AMANN
 

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 Perona-Malik equation of image processing.
Dedicated to S.M. Nikols'kii on the occasion of his 100th
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) = u0
, (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
tu - · a(u) u = f(u) on × (0, T),

  

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

 

Collections: Mathematics