The dynamics are studied for the Keller-Segel's minimal chemotaxis model
$τ u_t=(u_x-kuv_x)_x, \ \ \ \ v_t=v_{xx}-v+u$
on a bounded interval with homogeneous Neumann boundary conditions, where $\tau\geqslant 0$ and $k\gg 1$ are parameters and the total mass of $u$ is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which $u$ quickly becomes a delta like function with mass concentrated near the point of global maximum of $v$; in the second stage, the point of the global maximum of $v$ drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter $\tau$ is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.
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Figure 1.
A numerical solution of (1) with
Figure 2.
First Row: a numerical solution of (6) with
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