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A counterexample to finite time stopping property for one-harmonic map flow

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  • For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper.
    Mathematics Subject Classification: Primary: 35K67; Secondary: 35K51, 35K92, 58E20.


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