The regularity of the positive part of functions in L-2 (I; H-1(Omega)) boolean AND H-1(I; H-1(Omega)*) with applications to parabolic equations

Let u is an element of L-2(I; H-1(Omega)) with partial derivative(t)u is an element of 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 derivative(t)u+ is not an element of 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.