Well-posedness, blow-up dynamics and controllability of the classical chemotaxis model

Newton Okposo; Robert Willie

doi:10.1515/apam-2017-0122

Article

Well-posedness, blow-up dynamics and controllability of the classical chemotaxis model

Newton Okposo and Robert Willie

Published/Copyright: January 26, 2018

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From the journal Advances in Pure and Applied Mathematics Volume 10 Issue 2

Abstract

In this paper, we study well-posedness, existence of a lower finite time blow-up bound and variants of controllability of the classical chemotaxis model in Ω×(0,T), where Ω⊂ℝN, N=1,2,3. The spatial domain restrictions allow the system with initial data in L2⁢(Ω) to admit a solution in

L∞⁢[0,T;L2⁢(Ω))∩L2⁢(0,T;H1⁢(Ω))

and to have the property that the gradient chemical solutions are uniformly bounded in Ω×(0,T). A lower finite time blow-up bound of solutions in the norm of L2⁢(Ω) is proved using the differential inequality technique. Furthermore, using Carleman estimates and appropriate energy functionals, we show that the model is null and approximate controllable at any finite time T>0 with a single control in L2⁢(ω×(0,T)) acting on the cell-density equation, linearized through a priori uniform boundedness of the chemical drift solutions, where ω⊂Ω is a non-empty open subset of Ω. Lastly, bang-bang-type controls for the problem are constructed.

Keywords: Well-posedness; Keller–Segel model; lower finite time blow-up bound; null-approximate controllability; Carleman estimates; bang-bang controls

MSC 2010: 35B20; 35B40; 35B44; 35B65; 35K55; 35K57

Funding statement: This work was completed with the support of the University of KwaZulu-Natal, Sabbatical Leave Grant 2017–2018, for the first author. The second author was supported by Tetfund-Delta State University 2014–2017, PhD. Scholarship.

Acknowledgements

The second author is grateful to Prof. A. Rodríguez-Bernal for providing the reference [25, Theorem 6.1 p. 102], and to Prof. T. Dlotko for his invitation to visit the Institute of Mathematics at the University of Silesia in Katowice, Poland. He is sincerely grateful to the institute for its hospitality, and for fruitful discussions on results of this paper with Prof. T. Dlotko.

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Received: 2017-11-03

Accepted: 2017-12-25

Published Online: 2018-01-26

Published in Print: 2019-04-01

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Articles in the same Issue

https://doi.org/10.1515/apam-2017-0122

Keywords for this article

Well-posedness; Keller–Segel model; lower finite time blow-up bound; null-approximate controllability; Carleman estimates; bang-bang controls