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Summary: QUASILINEAR PARABOLIC FUNCTIONAL EVOLUTION
EQUATIONS
HERBERT AMANN
Institut f˜ur Mathematik, Universit˜at Z˜urich, Winterthurerstr. 190, CH--8057
Z˜urich, Switzerland, email: herbert.amann@math.unizh.ch
Based on our recent work on quasilinear parabolic evolution equations and maximal
regularity we prove a general result for quasilinear evolution equations with mem
ory. It is then applied to the study of quasilinear parabolic di#erential equations in
weak settings. We prove that they generate Lipschitz semiflows on natural history
spaces. The new feature is that delays can occur in the highest order nonlinear
terms. The general theorems are illustrated by a number of model problems.
Keywords: nonlinear evolution equations with memory, time delays, parabolic
functional di#erential equations, Volterra evolution equations.
Categories: 35R10, 35K90, 45K05: 45D05, 34K99
1. Introduction
In a recent paper [8] we have derived very general existence, uniqueness, and
continuity theorems for abstract quasilinear evolution equations of the form
—
u +A(u)u = F (u). (1)
Here A(u) is for each given u in an appropriate class of functions a bounded
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