Existence and asymptotic behaviour of solutions of the very fast diffusion equation

2011/09/16 by Hsu, Shu-Yu
#35B40 Secondary 35K65 #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 35K15

paper · doi:10.48550/arxiv.1109.3618

Abstract

Let n>2, 0max(1,(1-m)n/2), and 0≤ u₀∈ L_loc^p(Rⁿ) satisfy \liminf_R→∞R^-n+(2)/(1-m)∫_{|x|≤ R}u₀ dx=∞. We prove the existence of unique global classical solution of u_t=(n-1)/(m)Δu^m, u>0, in Rⁿ× (0,∞), u(x,0)=u₀(x) in \Rⁿ. If in addition 00, q2, if g_ij=u^(4)/(n+2)δ_ij is a metric on Rⁿ that evolves by the Yamabe flow ∂ g_ij/∂ t=-Rg_ij with u(x,0)=u₀(x) in Rⁿ where R is the scalar curvature, then u(x,t) is a global solution of the above fast diffusion equation.

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