摘要
We consider long time behavior of a given smooth convex embedded closed curve gamma 0 subset BbbR 2 evolving according to a nonlocal curvature flow, which arises in a Hele--Shaw problem and has a prescribed rate of change in its enclosed area A(t), i.e., dA/dt = - eta , where eta in ( - infty, infty ). Specifically, when the enclosed area expands at any fixed rate, i.e., eta in ( - infty, 0), or decreases at a fixed rate eta in (0, 2pi ), one has the round circle as the unique asymptotic shape of the evolving curves, while for a sufficiently large rate of area decrease, one can have n-fold symmetric curves (which look like regular polygons with smooth corners) as extinction shapes (self-similar solutions).