In dimensions n ≥ 3, radially symmetric solutions to the parabolic-elliptic Keller-Segel system form a Dirac mass at the origin continuously via a minimal measure-valued solution, with the singular mass θ(t) increasing strictly and absorbing the entire mass asymptotically.
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For the given chemotaxis model, uniform persistence holds when m ≥ 1; the positive equilibrium is linearly stable for χ0 below a parameter-dependent threshold χ*(u*) and unstable above it, with exponential convergence under stated conditions.
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Formation and Behavior of Dirac Singularities in the Parabolic-Elliptic Keller-Segel System in Dimensions $n\geq 3$
In dimensions n ≥ 3, radially symmetric solutions to the parabolic-elliptic Keller-Segel system form a Dirac mass at the origin continuously via a minimal measure-valued solution, with the singular mass θ(t) increasing strictly and absorbing the entire mass asymptotically.
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Chemotaxis models with signal-dependent sensitivity and a logistic-type source, II: Persistence and stabilization
For the given chemotaxis model, uniform persistence holds when m ≥ 1; the positive equilibrium is linearly stable for χ0 below a parameter-dependent threshold χ*(u*) and unstable above it, with exponential convergence under stated conditions.