• Journal of the Korean Mathematical Society
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• v.57 no.5
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• pp.1299-1322
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• 2020

We consider closed, star-shaped, admissible hypersurfaces in ℝⁿ⁺¹ expanding along the flow Ẋ = |X|^α-1 F^-β, α ≤ 1, β > 0, and prove that for the case α ≤ 1, β > 0, α + β ≤ 2, this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case α ≤ 1, α + β > 2, if furthermore the initial closed hypersurface is strictly convex, then the strict convexity is preserved during the evolution process and the flow blows up at finite time.