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When looking at a (nonlinear degenerate) PDE like

one defines as weak solution as

Now I wonder about (2.1). The deduction of (2.2) is ok to me, as I can use the standard way with partial integration of the original PDE. But I can't seem to find the deduction / motivation of (2.1).

Note: I did omit a lot of details here, as I hope they are not neccessary (in the way, that the deduction is kind of standard / generalized). The screenshots are from "Homogenization of a nonlinear degenerate parabolic differential equation" by Cao & Yue (2013).

blekkson3
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  • The references sends you to "Quasilinear Elliptic-parabolic Differential Equations" by H.W. Aft, S. Luckhaus. From what I understood, it seems (2.1) is meant to ensure an energy estimate? I would love to hear a better explanation from somebody that may have a better explanation though. – Rem Feb 12 '21 at 09:52

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