Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < \infty\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times \Omega).$$

We know that there exists a function $u \in L^2(0,T;H^1)\cap H^1(0,T;L^2(\Omega))$ satisfying $u(0) = u_0$ given $u_0 \in L^2(\Omega)$ such that $$\int_0^T \int_\Omega u'(t) \varphi(t) + \int_0^T \int_\Omega g(t)\nabla u(t) \nabla \varphi(t) = \int_0^T\int_\Omega f(t)\varphi(t)$$ for all $\varphi \in L^2(0,T;H^1(\Omega))$.

My question is, if we know that $u_0 \in L^\infty(\Omega)$, does it follow that $u' \in L^\infty((0,T) \times \Omega)$? I.e. do we have the expected regularity of $u$? We expect this because $f$ is appropriately regular and as is $u_0$ so we expect the same for $u'$.

If so can anyone refer me to a citation for this result? Thank you. If not, what additional smoothness do I need? I posted this on MSE too but did not receive any answers.

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