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# Dynamics of spike in a Keller-Segel's minimal chemotaxis model

• * Corresponding author: Yajing Zhang
This work is partially supported by NSF DMS-1008905, NNSFC (No. 61374089), China Scholarship Council, NSF of Shanxi Province(No. 2014011005-2), Hundred Talent Program of Shanxi and International Cooperation Projects of Shanxi Province (No. 2014081026).
• 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.

Mathematics Subject Classification: Primary:35B40, 92C17; Secondary: 92D15.

 Citation: • • Figure 1.  A numerical solution of (1) with $\tau=0$, $k=200$, $\ell=1$, and mesh sizes $\Delta x=1/400$, $\Delta t=3\times 10^{-6}$. Left: snapshots of $v(k,\cdot,t)$ with constant frequency; middle: snapshots of $u$; right: location of the point of maximum of $v(k,\cdot,t)$, which reaches the boundary at $T\approx 0.13$.

Figure 2.  First Row: a numerical solution of (6) with $k=100$, $\ell=4$, and $v_0(x)=\frac{\ell}{\pi}\big|\cos\frac{\pi(x-2.5)}{\ell}\big|$; left is location of maximum of $v$; middle is snapshots of $v(k,\cdot,t)$ with non-uniform time elapses; right is snapshots of $v_x(k,\cdot,t)$. Second Row: the solution of the free boundary problem (7) for $t\in[0,T^-]$ ($T\approx 5.2$), combined with the solution of the boundary value problem (8) for $t\geqslant T$; left is the location of free boundary; middle is snapshots of $w$; right is snapshots of $w_x$.

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