I am currently reading the paper [1]. The paper deals with the BVP: \begin{align*} \partial_t u - \Delta u + u(u-a) & = -qu \text{ in } ]0,T\mathclose[ \times \Omega \\[6pt] \partial_{n}u & = 0 \text{ in } ]0,T\mathclose[ \times \partial \Omega\\[6pt] u(0) & = u_0 \text{ in } \{0\} \times \Omega \end{align*}

where $u$ is the state, $q$ is the control and $u=S(q)$ the sol. operator.

To derive second-order optimality conditions, the authors calculate second-order derivatives $S''(q)[p,p]$, as the solution $v_{pp}$ to the BVP: \begin{align*} \partial_{t} v_{pp} - \Delta v_{pp} + (2u + q - a)\cdot v_{pp} & = -2v_p(v_p+p)\text{ in } \Omega_T \\[6pt] \partial_n v_{pp} & = 0 \text{ in } \Sigma_T \\[6pt] v_{pp}(0) & = 0 \text{ in } \Omega \end{align*} (Here $v_{p}=S'(q)(p)$)

And they reference [2]. Unfortunately, in [2], Thm. 5.16. the authors don't deal with mixed terms, like $-q\cdot u$. They consider only problems of the type: \begin{align*} \partial_t u - \Delta u +d(x,t,u) & = q \text{ in } \Omega_T\\[6pt] \partial_n u & = 0 \text{ in } \Sigma_T \\[6pt] u(0) & = u_0 \text{ in } \Omega \end{align*} That's why this is not directly applicable. Also the form of the BVP does not match the result of Thm. 5.16. So I am having the feeling that something else was used. I would be very grateful for any help or Ansatz to handle this.

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.

[2] F. Troeltzsch. Optimale Steuerung partieller Differentialgleichungen. Vieweg-Teubner, 2. ed., 2009