Delayed blow‐up for chemotaxis models with local sensing
Martin Burger, Philippe Laurençot, Ariane Trescases
Abstract
The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller–Segel model. The model we study has the same entropy as the minimal Keller–Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same; however, we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point of view via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller–Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the ( H 1 ) ′ -norm of the solutions, which can only grow with a square-root law in time. This additional ( H 1 ) ′ -bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.