Abstract

In this paper, we study the following parabolic chemo-repulsion with nonlinear production model in 2 D domains: {∂tu−Δu=∇⋅(u∇v),∂tv−Δv+v=up+fv1Ωc, with for 1 < p≤ 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.