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

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.

• As usual, it is not so easy to find readable references for parabolic equations. Krylov's book "Lectures on Elliptic and parabolic equations in Sobolev spaces" contains all ingredients to write dowon a proof... but deals only with parabolic equations in the whole space. The classical book by Ladyzenskaja-Solonnikov-Uralceva contains much more...the same for the book by Liebermann, if you are able to find the results (sometimes I do not find even the results I can prove). If you know the results when $g=0$ and full estimates in $W^{1,2}$, then the results follows from the method of continuity. Commented Jan 8 at 20:52
• I am sorry I cannot give a reference...I could explain how to prove it, but discussing. Maybe somebody else knows an easy reference. Commented Jan 8 at 20:53
• I don't get why AMS does not release the (very important) book by Ladyzenskaja-Solonnikov-Uralceva with modern LaTex typography. This book is so hard to read. Commented Jan 8 at 20:56
• @akira You are right, I find unreadable mainly the fonts. Commented Jan 8 at 21:00
• It seems Dautray-Lions Vol. 5, Chap. XV, §1 also contains results very close to what the OP is asking. Commented Jan 9 at 7:22

• 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.