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

Authors
  • Okposo, Newton
  • Willie, Robert
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Jan 26, 2018
Volume
10
Issue
2
Pages
93–123
Identifiers
DOI: 10.1515/apam-2017-0122
Source
De Gruyter
Keywords
License
Yellow

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 ) {\Omega\times(0,T)} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} , N = 1 , 2 , 3 {N=1,2,3} . The spatial domain restrictions allow the system with initial data in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} to admit a solution in L ∞ ⁢ [ 0 , T ; L 2 ⁢ ( Ω ) ) ∩ L 2 ⁢ ( 0 , T ; H 1 ⁢ ( Ω ) ) L^{\infty}[0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega)) and to have the property that the gradient chemical solutions are uniformly bounded in Ω × ( 0 , T ) {\Omega\times(0,T)} . A lower finite time blow-up bound of solutions in the norm of L 2 ⁢ ( Ω ) {L^{2}(\Omega)} 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 {T>0} with a single control in L 2 ⁢ ( ω × ( 0 , T ) ) {L^{2}(\omega\times(0,T))} acting on the cell-density equation, linearized through a priori uniform boundedness of the chemical drift solutions, where ω ⊂ Ω {\omega\subset\Omega} is a non-empty open subset of Ω. Lastly, bang-bang-type controls for the problem are constructed.

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