In this paper, we study the following parabolic chemo-repulsion with nonlinear production model in \(2D\) domains: $$ \left \{ \textstyle\begin{array}{rcl} \partial _{t}u-\Delta u&=&\nabla \cdot (u\nabla v), \\ \partial _{t}v-\Delta v+v&=&u^{p}+fv\, 1_{\varOmega _{c}}, \end{array}\displaystyle \right . $$ with for \(1< p\leq 2\). This system is related to a bilinear control problem, where the state \((u,v)\) is the cell density and the chemical concentration respectively, and the control \(f\) acts in a bilinear form in the chemical equation. We prove the existence and uniqueness of global-in-time strong state solution for each control, and the existence of global optimum solution. Afterwards, we deduce the optimality system for any local optimum via a Lagrange multipliers theorem, proving extra regularity of the Lagrange multipliers. The case \(p>2\) remains open.

中文翻译：

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具有非线性生产的二维化学排斥模型及相关的最优控制问题

在本文中，我们使用\（2D \）域中的非线性生产模型研究以下抛物线化学排斥：$$ \ left \ {\ textstyle \ begin {array} {rcl} \ partial _ {t} u- \ Delta u＆=＆\ nabla \ cdot（u \ nabla v），\\ \ partial _ {t} v- \ Delta v + v＆=＆u ^ {p} + fv \，1 _ {\ varOmega _ {c}}，\ end {array} \ displaystyle \ right。$$与\（1 <p \ leq 2 \）。该系统与双线性控制问题有关，其中状态\（（u，v）\）分别是细胞密度和化学浓度，而控制\（f \）在化学方程式中以双线性形式起作用。我们证明了每个控制的全局及时强状态解的存在和唯一性，以及全局最优解的存在。之后，我们通过拉格朗日乘数定理推论出任何局部最优的最优系统，证明拉格朗日乘数的额外规律性。\（p> 2 \）保持打开的情况。