Exact minimum speed of traveling waves in a Keller–Segel model

The paper proves that, for the 1D parabolic–parabolic Keller–Segel system with logistic source and 0 ≤ χ(v) ≤ μ on [0,β^{-1}], nonnegative, monotone traveling waves connecting (1,β^{-1}) to (0,0) exist if and only if c ≥ 2 max{√μ, √(Dβ)}, are unique (up to translation), and satisfy 0 ≤ U ≤ βV ≤ 1; it does so via a spatial-dynamical trapping-region construction and eigenvalue analysis at the leading edge . The candidate solution derives the same threshold and qualitative properties, but by a different route: an integrating-factor identity for V yielding 0 ≤ U ≤ βV and V' ≤ 0, a linearized leading-edge argument giving the same necessary condition c ≥ 2 max{√μ, √(Dβ)}, and a homotopy (in chemotactic strength p) plus Leray–Schauder continuation for sufficiency, with a sliding method for uniqueness. Thus, the statements align; the proofs are substantively different. The paper’s results (Theorem 2) match the model’s conclusions exactly, including minimal speed, monotonicity, ordering, and uniqueness .