Unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus model on the unit disk

We identify unstable radial equilibria as sharp concentration thresholds for the Jäger--Luckhaus chemotaxis system on the unit disk with no-flux boundary conditions. Concentration is compared through cumulative mass. At fixed total mass, every radial initial profile more concentrated than an unstable equilibrium produces finite-time blow-up, whereas every nonstationary profile less concentrated than that equilibrium generates a global, uniformly bounded solution converging to the least concentrated radial equilibrium. We also classify the radial equilibria and determine their nonlinear stability. Let $Λ= j_{1,1}^{2}$, where $j_{1,1}$ is the first positive zero of the Bessel function $J_1$. For $M\ne Λπ$, the least concentrated equilibrium is the unique locally asymptotically stable equilibrium under mass-preserving radial perturbations. It is spatially homogeneous for $M<Λπ$ and strictly increasing for $M>Λπ$. Every other radial equilibrium, including the homogeneous state at $M=Λπ$, is nonlinearly unstable. These results clarify how total population and initial spatial aggregation jointly determine collapse or bounded redistribution, and how stable peripheral enrichment can occur within the radial model.