A lower bound on the blow up time for solutions of a chemotaxis system with nonlinear chemotactic sensitivity

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+χu(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 Rn 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.