| Abstract: |
| We consider radially symmetric solutions to a degenerate parabolic--elliptic Keller--Segel system in bounded balls with initial data having compact support. Our main result shows that the initial evolution of the positivity set is essentially completely determined by the flatness/steepness of the initial data near a boundary point $x_0$ of the support. If they are sufficiently flat (respectively, steep), the support shrinks (respectively, expands) near $x_0$. We give concrete conditions for both behaviors and in particular show that there is a critical exponent and a critical parameter distinguishing between these cases. The proof is based on constructing suitable sub- and supersolutions to a transformed problem. |
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