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