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We consider the Keller-Segel system of degenerate type (KS)m with m > 1 below. We establish a uniform estimate of ∂2 xum−1 from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)m. In addition, we find that the set D(t) := {x ∈ R;u(x, t) > 0} of positive region to the solution u is monotonically non-decreasing with respect to t.

Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems