Existence and properties of ancient solutions of the Yamabe flow

2016/06/09 by Hsu, Shu-Yu
#35A01 #35B44 (Secondary) #35K55 #53C44 (Primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.02914

Abstract

Let n≥ 3 and m=(n-2)/(n+2). We construct 5-parameters, 4-parameters, 3-parameters ancient solutions of the equation v_t=(v^m)_xx+v-v^m, v>0, in ℝ× (-∞,T) for some T∈ℝ. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as |x|→∞. We also prove that both the 3-parameters ancient solution and the 4-parameters ancient solution are singular limit solution of the 5-parameters ancient solutions.

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