A lower bound on the blow up time for solutions of a chemotaxis system with nonlinear chemotactic sensitivity. (August 2017) - Anderson, Jeffrey R.; Deng, Keng
Abstract: In a recent study, a lower bound is established on the blow up time for solutions of a chemotaxis system, with nonlinear chemotactic sensitivity u ( u + 1 ) m − 1, set in the three-dimensional unit ball. Here, u is the density of a cell or organism that produces a chemical, with density v, and moves preferentially toward regions of higher concentration of v according to the flux − ∇ u + χ u ( u + 1 ) m − 1 ∇ v . With χ > 0, v is referred to as a "chemoattractant" and, in the case m = 1, the system reduces to a version of the Keller–Segel model. Solutions that blow up in finite time have been previously established for the system on a ball in R n provided n ≥ 2, m > 2 / n . For technical reasons, the lower bound proven for the blow up time applies in such cases when n = 3 and m ≤ 2 . We extend the analysis and resulting lower bound to such a model in general convex domains, with n ≥ 2 and any m .