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Bochner integrable function; projection onto non-negative functions; parabolic equation
Let $u\in L^2(I; H^1(\Omega))$ with $\partial_t u\in L^2(I; H^1(\Omega)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \notin L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.
[1] Gajewski H., Gröger K., Zacharias K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974. MR 0636412
[2] Grün G.: Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening. Z. Anal. Anwendungen 14 (1995), no. 3, 541–574. DOI 10.4171/ZAA/639 | MR 1362530
[3] Roubíček T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, 153, Birkhäuser, Basel, 2013. MR 3014456 | Zbl 1270.35005
[4] J. Wloka J.: Partielle Differentialgleichungen. Teubner, Stuttgart, 1982. MR 0652934 | Zbl 0482.35001
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