On a bi-dimensional chemo-repulsion model with nonlinear production

Abstract

In this paper, we study the following parabolic chemo-repulsion with nonlinear production model: $$\{\begin{array}{rcl}{\partial }_{t}u-\Delta u& =& \nabla \cdot (u\nabla v),\\ {\partial }_{t}v-\Delta v+v& =& u^p+fv{1}_{{\Omega }_c}.\end{array}$$ 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 $1<p<2$.

References in corpus (3)

• Study of a chemo-repulsion model with quadratic production. Part I: Analysis of the continuous problem and time-discrete numerical schemes
• Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes
• Study of a chemo-repulsion model with quadratic production. Part II: Analysis of an unconditional energy-stable fully discrete scheme