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

- Authors
- 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.