2020/09/23 by Vincent Calvez, Franca Hoffmann, Calvez, Vincent +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #35A23 #35B40 #Analysis of PDEs (math.AP) #Cell Image Analysis Techniques #FOS: Mathematics #Mathematical Biology Tumor Growth #Medical Image Segmentation Techniques
paper · doi:10.48550/arxiv.2009.11048
openalex publication_date 2020/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results.