Is \int_{0}^{\tau}\int_{\Omega}{(\partial_{t} u_{h})\psi}\geq 0 where \psi=\theta(t) \frac{u^{q}(x,t+h)-u^{q}(x,t)}{h}
where u^{h}(\cdot, t)=\frac{1}{h}\int_{0}^{t}{e^{(s-t)/h}u(\cdot, s)ds} and \theta is an approximation of the characteristic function, vanishing on the boundary? I want to assume that u\in C\left(I; L^{1+q}(\Omega)\right)\quad \textnormal{and} u^{q}\in L_{loc}^{p}\left(I; W^{1, p}(\Omega)\right).
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