| Abstract: |
| We study chemotaxis effect (chi) vs logistic damping (mu) on boundedness (and large time behavior) for the minimal Keller-Segel model with logistic source in 2- and 3-D smooth and bounded domains. We obtain qualitative boundedness on chi and mu: up to a scaling constant depending only on initial data and the underlying domain, we provide explicit upper bounds for the solution components of the corresponding initial-boundary value problem. These bounds are increasing in chi and decreasing in mu.
In 2-D, the corresponding upper bounds have only one singularity in mu at mu=0. In contrast, in 3-D, the upper bounds, holding under a critical explicit relation between chi and mu (which has been shown to guarantee boundedness ), are defined for all chi and mu>const. chi, and, have two singularities in mu at mu=0 and mu=const. chi. It is worthwhile to mention that, in the absence of logistic source, the corresponding classical KS model is well-known to possess blow-ups for even small initial data. We hope that these qualitative findings presented here would produce some new principles on finite-time blow-up to chemotaxis systems with weak logistic damping sources. |
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