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Boundary layers and stabilization of the singular Keller-Segel system

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  • The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε→0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-ε estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L1-estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

    Mathematics Subject Classification: Primary: 92C17, 35Q92, 35K57; Secondary: 35A01, 35B40, 35B44.

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  • Figure 1.  Numerical simulation of the evolution of solution profiles of the system (4) as $\varepsilon\to 0$ in the interval $[0, 1]$, where $u|_{x=0, 1}=1+0.1\sin(t), v|_{x=0, 1}=1+0.1\sin(t), u_0(x)=1-\sin(\pi x), v_0(x)=1+x(1-x)$. The solution $(u(x,t), v(x,t)$ is plotted at time $t=0.2$

    Figure 2.  Numerical simulation of the time evolution of boundary layer solutions of (4) with $\varepsilon=0.0001$ in the interval $[0, 1]$, where the initial and boundary date are same as those chosen in Fig. 1

    Figure 3.  Numerical simulation of the time evolution of solutions to (4) in the interval $[0, 1]$ with decay boundary data, where $u|_{x=0,1}=1+\exp(-t), v|_{x=0, 1}=1/(1+t), u_0(x)=2+x(1-x),v_0(x)=1+x(1-x)$, and $\chi=D=1, \varepsilon=0.0001$

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