I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ \nabla y \cdot n = 0 \text{ in } ]0,T[ \times \partial \Omega\\ y(0) = y_0 \text{ in } \Omega \end{align*}

with $f \in L^2(Q), g \in L^{\infty}(Q)$, $y_0 \in H^1(\Omega)$.

The authors then state that there exists a solution $y \in \mathcal{C}(0,T;V)$ such that:
\begin{equation*}
\lVert y \rVert_{\mathcal{C}(0,T;H^1)} \leq C \cdot (\lVert f \rVert_{L^2(Q)} + \lVert y_0 \rVert_{H^1(\Omega)})
\end{equation*}

is there some reference to this ? I can't get this to work just using the Galerkin approximation. Thanks in advance !

**Reference**

[1] Malte Braack, Martin F. Quaas, Benjamin Tews, Boris Vexler, "Optimization of fishing strategies in space and time as a non-convex optimal control problem", (English) Journal of Optimization Theory and Applications 178, No. 3, 950-972 (2018), DOI:10.1007/s10957-018-1304-7, MR3836262, Zbl 1409.49018.