Singular limit of solutions of the very fast diffusion equation

We prove that the distribution solutions of the very fast diffusion equation ∂u/∂t=Δ(um/m)∂u/∂t=Δ(um/m), u>0u>0, in Rn×(0,∞)Rn×(0,∞), u(x,0)=u0(x)u(x,0)=u0(x) in RnRn, where m<0m<0, n≥2n≥2, constructed in [P. Daskalopoulos, M.A. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal. 137 (1997) 363–380] are actually classical maximal solutions of the problem. Under the additional assumption that u0⁄∈L1(Rn)u0⁄∈L1(Rn), 0≤u0∈Llocp(Rn) for some constant p>n/2p>n/2, and u0(x)≥ε/|x|2αu0(x)≥ε/|x|2α for any |x|≥R1|x|≥R1 where ε>0ε>0, R1>0R1>0, m0<0m0<0, α(1−m0)n/2p>(1−m0)n/2, we prove that the solution of the above problem will converge uniformly on every compact subset of Rn×(0,∞)Rn×(0,∞) to the maximal solution of the equation vt=Δlogvvt=Δlogv, v(x,0)=u0(x)v(x,0)=u0(x), as m↗0−m↗0−. For any smooth bounded domain Ω⊂RnΩ⊂Rn, m0<0m0<0, m∈[m0,0)∪(0,1)m∈[m0,0)∪(0,1), and 0≤u0∈Lp(Ω)0≤u0∈Lp(Ω) for some constant p>(1−m0)max(1,n/2)p>(1−m0)max(1,n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem ∂u/∂t=Δ(um/m)∂u/∂t=Δ(um/m), u>0u>0, in Ω×(0,∞)Ω×(0,∞), u=u0u=u0 in ΩΩ, u=gu=g on ∂Ω×(0,∞)∂Ω×(0,∞) with either finite or infinite positive boundary value gg. We also prove a similar convergence result for the solutions of the above Dirichlet problem as m→0m→0.