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

Let $u\in L^2(I;H^1(\mathrm{\Omega }))$ with ${\mathrm{\partial }}_{t}u\in L^2(I;H^1(\mathrm{\Omega }{)}^{\ast })$ 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 $\partial_t u^+ \not\in L^1(I; H^1(\Omega)^*)$ingeneral\mathrm{.}Nevertheless,$u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.