Abstract
This is a survey of the most common approaches to quasi-linear parabolic evolution equations, a discussion of their advantages and drawbacks, and a presentation of an entirely new approach based on maximal regularity. The general results here apply, above all, to parabolic initial-boundary value problems that are non-local in time. This is illustrated by indicating their relevance for quasi-linear parabolic equations with memory and, in particular, for time-regularized versions of the Perona-Malik equation of image processing.
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