Abstract
In this paper, we study the following parabolic chemo-repulsion with nonlinear production model in \(2D\) domains:
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.
Similar content being viewed by others
References
De Araujo, A.L.A., De Magalhães, P.M.D.: Existence of solutions and optimal control for a model of tissue invasion by solid tumours. J. Math. Anal. Appl. 421, 842–877 (2015)
Chaplain, M.A., Lolas, G.: Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734 (2005)
Chaplain, M.A., Stuart, A.: A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168 (1993)
Cieślak, T., Laurençot, P., Morales-Rodrigo, C.: Global existence and convergence to steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1. In: Banach Center Publ. vol. 81, pp. 105–117. Polish Acad. Sci. Inst. Math., Warsaw (2008)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel (2009)
Fister, K.R., Mccarthy, C.M.: Optimal control of a chemotaxis system. Q. Appl. Math. 61, 193–211 (2003)
Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A.: Study of a chemo-repulsion model with quadratic production. Part I: analysis of the continuous problem and time-discrete numerical schemes. Comput. Math. Appl. 80(5), 692–713 (2020)
Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A.: Study of a chemo-repulsion model with quadratic production. Part II: analysis of an unconditional energy-stable fully discrete scheme. Comput. Math. Appl. 80(5), 636–652 (2020)
Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A.: Analysis of a chemo-repulsion model with nonlinear production: the continuous problem and unconditionally energy stable fully discrete schemes. Submitted. arXiv:1807.05078v2
Guillén-González, F., Mallea-Zepeda, E., Rodríguez-Bellido, M.A.: Optimal bilinear control problem related to a chemo-repulsion system in 2D domains. ESAIM Control Optim. Calc. Var. 26, paper No. 29, 21pp (2020)
Guillén-González, F., Mallea-Zepeda, E., Rodríguez-Bellido, M.A.: A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem. SIAM J. Control Optim. 58(3), 1457–1490 (2020)
Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)
Maini, P.K., Myerscough, M.R., Winters, K.H., Murray, J.D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53, 701–719 (1991)
Mantzaris, N.V., Webb, S., Othmer, H.G.: Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49, 111–187 (2004)
Myerscough, M.R., Maini, P.K., Painter, K.J.: Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60, 1–26 (1998)
Necas, L.: Les méthodes Directes en Théorie des Equations Elliptiques. Editeurs Academia, Prague (1967)
Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3). 13, 115–162 (1959)
Rodríguez-Bellido, M.A., Rueda-Gómez, D.A., Villamizar-Roa, E.J.: On a distributed control problem for a coupled chemotaxis-fluid model. Discrete Contin. Dyn. Syst., Ser. B 23, 557–571 (2018)
Ryu, S.-U., Yagi, A.: Optimal control of Keller-Segel equations. J. Math. Anal. Appl. 256, 45–66 (2001)
Ryu, S.U.: Optimal control for a parabolic system modelling chemotaxis. Trends Math. 6, 45–49 (2003)
Winkler, M.: A critical blow-up exponent in a chemotaxis system with nonlinear signal production. Nonlinearity 31, 2031–2056 (2018)
Tao, Y.: Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin. Dyn. Syst., Ser. B 18, 2705–2722 (2013)
Tindall, M.J., Maini, P.K., Porter, S.L., Armitage, J.P.: Overview of mathematical approaches used to model bacterial chemotaxis. II. Bacterial populations. Bull. Math. Biol. 70, 1570–1607 (2008)
Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications. AMS, Providence (2010)
Tyson, R., Lubkin, S.R., Murray, J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38, 359–375 (1999)
Woodward, D., Tyson, R., Myerscough, M., Murray, J.D., Budrene, E., Berg, H.: Spatio-temporal patterns generated by salmonella typhimurium. Biophys. J. 68, 2181–2189 (1995)
Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 6, 49–62 (1979)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-020-00365-3