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The aim of this paper is to analyze a model for chemotaxis based on a localsensing mechanism instead of the gradient sensing mechanism used in thecelebrated minimal Keller-Segel model. The model we study has the same entropyas the minimal Keller-Segel model, but a different dynamics to minimize thisentropy. Consequently, the conditions on the mass for the existence ofstationary solutions or blow-up are the same, however we make the interestingobservation that with the local sensing mechanism the blow-up in the case ofsupercritical mass is delayed to infinite time. Our observation is maderigorous from a mathematical point via a proof of global existence of weaksolutions for arbitrary large masses and space dimension. The key difference ofour model to the minimal Keller-Segel model is that the structure of theequation 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 intime. This additional $(H^1)'$-bound implies a lower bound on the entropy,which contrasts markedly with the minimal Keller-Segel model for which it isunbounded from below in the supercritical case. Besides, regularity anduniqueness of solutions are also studied.

Delayed Blow-Up for Chemotaxis Models with Local Sensing