On a Bi-dimensional Chemo-repulsion Model with Nonlinear Production and a Related Optimal Control Problem
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On a Bi-dimensional Chemo-repulsion Model with Nonlinear Production and a Related Optimal Control Problem Francisco Guillén-González1 · Exequiel Mallea-Zepeda2 · Élder J. Villamizar-Roa3
Received: 22 March 2020 / Accepted: 24 September 2020 © Springer Nature B.V. 2020
Abstract In this paper, we study the following parabolic chemo-repulsion with nonlinear production model in 2D domains:
∂t u − u = ∇ · (u∇v), ∂t v − v + v = up + f v 1Ωc ,
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. Keywords Chemo-repulsion · Nonlinear production · Strong solutions · Uniqueness · Optimal control Mathematics Subject Classification (2010) 35Q35 · 35Q92 · 49J20 · 49K20
B E. Mallea-Zepeda [email protected]
F. Guillén-González [email protected] É.J. Villamizar-Roa [email protected] 1
Dpto. de Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad de Sevilla, C/Tarfia, S/N, 41012, Sevilla, Spain
2
Departamento de Matemática, Universidad de Tarapacá, Av. 18 de septiembre, 2222, Arica, Chile
3
Escuela de Matemáticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
F. Guillén-González et al.
1 Introduction In last years, the understanding of chemotaxis systems has proceeded to propose more elaborated models in order to capture certain types of mechanisms of interaction between the cells and the chemical substances, which advice to introduce different kinds of growth of substances by the action of cells. A particular case of those mechanisms, which is not properly captured by classical chemotaxis models, corresponds to the process of signal production through cells, which may depend on the population density in a nonlinear manner, as for instance, the saturation effects produced by some bacterial chemotaxis and the pattern formation [12] (see also [13, 15, 21, 23]). In this paper we are interested in the mathematical analysis of a bi-dimensional chemorepulsion model with nonlinear signal production. By chemo-repulsion we mean the biophysical process of the cell movement towards a lower concentration of chemical substance. This model is given by the following parabolic system in Q := Ω × (0, T ): ∂t u − u = ∇ · (u∇v), (1) ∂t v − v + v = up + f v 1Ωc . Here, Ω ⊂ R2 is a bounded domain and (0, T ) a time interval. The unknowns are u(t, x) ≥ 0 and v(t, x) ≥ 0 denoting the cell density and the chemical concentration, respectively. The term up , p > 1, is the nonlinear production and the reaction term f v 1Ωc can be interpreted as a bilinear control where the control f acts as a proliferation or de
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