论文标题
在具有非线性产生的双维化学抑制模型上
On a bi-dimensional chemo-repulsion model with nonlinear production
论文作者
论文摘要
在本文中，我们研究了以下非线性生产模型的以下抛物面化学抑制：$$ \ left \ {\ begin {array} {rcl} \ partial_tu-u-u- =＆=＆\ nabla \ cdot（u \ nabla v） 1_ {ω_c}。 \ end {array} \ right。 $$此问题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，并且控制$ f $在化学方程中以双线性形式起作用。对于$ 2D $域，我们首先考虑了二次信号生产的情况（$ p = 2 $），证明了每个控件的全球强状态解决方案的存在和唯一性，以及全球最佳解决方案的存在。之后，我们通过Lagrange乘数定理推导任何本地最佳的最佳系统，证明了Lagrange乘数的规律性。最后，我们考虑信号生产的情况$ u^p $，$ 1 <p <2 $。
In this paper, we study the following parabolic chemo-repulsion with nonlinear production model: $$ \left\{ \begin{array}{rcl} \partial_tu-Δu&=&\nabla\cdot(u\nabla v),\\ \partial_tv-Δv+v&=&u^p+fv\, 1_{Ω_c}. \end{array} \right. $$ This problem 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. For $2D$ domains, we first consider the case of quadratic signal production ($p=2$), proving the existence and uniqueness of global 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 multiplier Theorem, proving regularity of the Lagrange multipliers. Finally, we consider the case of signal production $u^p$ with $1<p<2$.
