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Delayed blow‐up for chemotaxis models with local sensing
- Source :
- Journal of the London Mathematical Society. 103:1596-1617
- Publication Year :
- 2020
- Publisher :
- Wiley, 2020.
-
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 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.
- Subjects :
- General Mathematics
010102 general mathematics
Space dimension
Mathematics::Analysis of PDEs
Structure (category theory)
Duality (optimization)
01 natural sciences
Upper and lower bounds
Supercritical fluid
Quantitative Biology::Cell Behavior
010101 applied mathematics
Entropy (classical thermodynamics)
Applied mathematics
Point (geometry)
Uniqueness
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14697750 and 00246107
- Volume :
- 103
- Database :
- OpenAIRE
- Journal :
- Journal of the London Mathematical Society
- Accession number :
- edsair.doi...........80bad8dce8f91b89aced5615fb0ced50