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The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper. 2010 Mathematics subject classification: primary 35K55; secondary 35B35, 35K15, 53C44.doi:10.1017/s0004972710001942 fatcat:vn5v4bj2erd7zj2lpxbu3t5n4i