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On a Bi-dimensional Chemo-repulsion Model with Nonlinear Production and a Related Optimal Control Problem

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Abstract

In this paper, we study the following parabolic chemo-repulsion with nonlinear production model in \(2D\) domains:

$$ \left \{ \textstyle\begin{array}{rcl} \partial _{t}u-\Delta u&=&\nabla \cdot (u\nabla v), \\ \partial _{t}v-\Delta v+v&=&u^{p}+fv\, 1_{\varOmega _{c}}, \end{array}\displaystyle \right . $$

with for \(1< p\leq 2\). 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.

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Acknowledgements

F. Guillén-González has been partially financed by the Project PGC2018-098308-B-I00, funded by FEDER / Ministerio de Ciencia e Innovación—Agencia Estatal de Investigación. E. Mallea-Zepeda has been supported by Proyecto UTA-Mayor 4751-20, Universidad de Tarapacá. E.J. Villamizar-Roa has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, proyecto de año sabático.

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Authors and Affiliations

  1. Dpto. de Ecuaciones Diferenciales y Análisis Numérico and IMUS, Universidad de Sevilla, C/Tarfia, S/N, 41012, Sevilla, Spain

    Francisco Guillén-González

  2. Departamento de Matemática, Universidad de Tarapacá, Av. 18 de septiembre, 2222, Arica, Chile

    Exequiel Mallea-Zepeda

  3. Escuela de Matemáticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia

    Élder J. Villamizar-Roa

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  1. Francisco Guillén-González
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  2. Exequiel Mallea-Zepeda
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  3. Élder J. Villamizar-Roa
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Correspondence to Exequiel Mallea-Zepeda.

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Guillén-González, F., Mallea-Zepeda, E. & Villamizar-Roa, É.J. On a Bi-dimensional Chemo-repulsion Model with Nonlinear Production and a Related Optimal Control Problem. Acta Appl Math 170, 963–979 (2020). https://doi.org/10.1007/s10440-020-00365-3

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