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On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model

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  • In this paper, we continue to study a general nonlocal gradient Patlak-Keller-Segel chemotaxis model in a one dimensional spatial domain. By utilizing the properties of the nonlocal gradient, we first apply the well-known Moser-Alikakos iteration technique plus the heat semigroup theory to obtain the boundedness and hence the global existence of its solution. Then we study the asymptotic behavior of the time-dependent solution, and obtain the limiting equations when the sampling radius $\rho\rightarrow 0$ as well as convergence results when time $t\rightarrow \infty$. Along this way, a ``global" stability issue of the spiky stationary solution for the minimal model is formulated. Finally and importantly, we study the stability of the nonconstant bifurcating solutions. Interestingly, the small size of the cells enhances the occurrence of pattern formation, the stability results are independent of the net creation rate of the chemical, and the stability is closely related to the cell radius $\rho$. Typically, when the cell (net) degradation rate lies below a threshold (stabilizing) value, the cell is stable. Surprisingly, this threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical signal, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.
    Mathematics Subject Classification: Primary: 35K57, 35B40, 92C17, 35B32; Secondary: 35K45, 35J67, 35B41, 35B36.


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