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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 $\partial_t u^+\not\in L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies anintegration-by-parts formula, which can be used to prove non-negativity of weaksolutions of parabolic equations.

The regularity of the positive part of functions in $L^2(I; H^1(\Omega)) \cap H^1(I; H^1(\Omega)^*)$ with applications to parabolic equations