  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
Poincar\'e 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.
