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Convergence to the grim reaper for a curvature flow with unbounded boundary slopes. (English) Zbl 1471.35190

Summary: We consider a curvature flow \(V=H\) in the band domain \(\Omega :=[-1,1]\times \mathbb{R} \), where, for a graphic curve \(\Gamma_t\), \(V\) denotes its normal velocity and \(H\) denotes its curvature. If \(\Gamma_t\) contacts the two boundaries \(\partial_\pm \Omega\) of \(\Omega\) with constant slopes, in 1993, S. J. Altschuler and L.-F. Wu [Math. Ann. 295, No. 4, 761–765 (1993; Zbl 0798.35063)] proved that \(\Gamma_t\) converges to a grim reaper contacting \(\partial_\pm \Omega\) with the same prescribed slopes. In this paper we consider the case where \(\Gamma_t\) contacts \(\partial_\pm \Omega\) with slopes equaling to \(\pm 1\) times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in \(C^{2,1}_{loc} ((-1,1)\times \mathbb{R})\) topology to the grim reaper with span \((-1,1)\).

MSC:

35K93 Quasilinear parabolic equations with mean curvature operator
35C07 Traveling wave solutions
35K20 Initial-boundary value problems for second-order parabolic equations
53E10 Flows related to mean curvature

Citations:

Zbl 0798.35063
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References:

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