A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models

Abstract

In the present review we focus on the chemotaxis-consumption model $$\partial _t u - \Delta u = - \nabla \cdot (u \nabla v)$$ ∂ t u - Δ u = - ∇ · ( u ∇ v ) and $$\partial _t v - \Delta v = - u^s v$$ ∂ t v - Δ v = - u s v in $$(0,T) \times \Omega $$ ( 0 , T ) × Ω , for any fixed $$s \ge 1$$ s ≥ 1 , endowed with isolated boundary conditions and nonnegative initial conditions, where (u, v) model cell density and chemical signal concentration. Our objective is to present an overview of the related literature and latest results on the aforementioned model concerning the following three distinct research lines we have obtained in Corrêa Vianna Filho and Guillén-González (Nonlinear Anal Real World Appl 70, 103795, 2023), Guillén-González and Corrêa Vianna Filho (SIAM J Numer Anal 61(5), 2509–2533, 2023), Guillén-González and Corrêa Vianna Filho (SIAM J Control Optim 61(5), 3156–3182, 2023), Corrêa Vianna Filho and Guillén-González (Appl Math Optim 89(2), 48, 2024): the mathematical analysis, the numerical analysis and the related optimal control theory with a bilinear control acting on the chemical equation.