Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$

Question

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.

Answer 1

I also do not know an easy and direct reference, in particular since there are homogeneous Neumann boundary conditions which are usually not treated in examples.

Anyway, here is a blueprint how one could argue, but I am not claiming that this is the minimal or most direct/elegant way. I am just giving a possible blueprint based on a nonautonomous perturbation result of maximal parabolic regularity as in [ACFP]. (See also there for notions and further references.)

• On a sufficiently smooth or convex domain $\Omega$, the domain of the (negative) Neumann Laplacian $-\Delta_N$ in $L^2(\Omega)$ is $H^2(\Omega)$. It also satisfies maximal parabolic regularity on $L^2(\Omega)$ since its negative generates an analytic semigroup.
• From [ACFP Proposition 1.3] it follows that the operator family $A(t) := -\Delta_N + g(t)$ still satisfies (now: nonautonomous) maximal parabolic regularity on $L^2(\Omega)$ with domain $H^2(\Omega)$.
• Thus, for every $f \in L^2(Q)$ and every $y_0 \in (L^2(\Omega),H^2(\Omega))_{1/2,2}$, the equation in OP has a unique solution $u \in \mathcal W := W^{1,2}(0,T;L^2(\Omega) \cap L^2(0,T;H^2(\Omega))$ with $$\|u\|_{\mathcal W} \lesssim \|f\|_{L^2(Q)} + \|y_0\|_{(L^2(\Omega),H^2(\Omega))_{1/2,2}}.$$
• It remains to identify $(L^2(\Omega),H^2(\Omega))_{1/2,2} = H^1(\Omega)$, e.g. by considering a continuous linear extension operator which takes the function spaces in question to their $\mathbb{R}^n$ version where the result is a standard one, and to recall the embedding $$\mathcal W \hookrightarrow C\bigl([0,T];(L^2(\Omega),H^2(\Omega))_{1/2,2}\bigr) = C([0,T];H^1(\Omega)).$$

In fact, I think that one could dispose of most, if not all, regularity assumptions on $\Omega$ by arguing in a more sharp manner, since the $H^2(\Omega)$ regularity is not explicitly asked for, but this seems not to be the focus of the question.

[ACFP] Arendt, Wolfgang; Chill, Ralph; Fornaro, Simona; Poupaud, César, $L^{p}$-maximal regularity for nonautonomous evolution equations, J. Differ. Equations 237, No. 1, 1-26 (2007). ZBL1126.34037.