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Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation

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  • We are concerned with a multiple boundary spike solution to the steady-state problem of a chemotaxis system: $P_t=\nabla \cdot \big( P\nabla ( \log \frac{P}{\Phi (W)})\big)$, $W_t=ε^2 \Delta W+F(P,W)$, in $\Omega \times (0,\infty)$, under the homogeneous Neumann boundary condition, where $\Omega\subset \mathbb{R}^N$ is a bounded domain with smooth boundary, $P(x,t)$ is a population density, $W(x,t)$ is a density of chemotaxis substance. We assume that $\Phi(W)=W^p$, $p>1$, and we are interested in the cases of $F(P,W)=-W+\frac{PW^q}{\alpha+\gamma W^q}$ and $F(P,W)=-W+\frac{P}{1+ k P}$ with $q>0, \alpha, \gamma, k\ge 0$, which has a saturating growth. Existence of a multiple spike stationary pattern is related to a weak saturation effect of $F(P,W)$ and the shape of the domain $\Omega$. In this paper, we assume that $\Omega$ is symmetric with respect to each hyperplane $\{ x_1=0\},\cdots ,\{ x_{N-1}=0\}$. For two classes of $F(P,W)$ above with saturation effect, we show the existence of multiple boundary spike stationary patterns on $\Omega$ under a weak saturation effect on parameters $\alpha,\gamma$ and $k$. Based on the method developed in [14] and [10], we shall present some technique to construct a multiple boundary spike solution to some reduced nonlocal problem on such domains systematically.
    Mathematics Subject Classification: Primary: 35K50, 35Q80; Secondary: 92C15.

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