We prove that the distribution solutions of the very fast diffusion equation âu/ât=Î(um/m), u>0, in RnĂ(0,â), u(x,0)=u0(x) in Rn, where m<0, nâ„2, constructed in [P. Daskalopoulos, M.A. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal. 137 (1997) 363â380] are actually classical maximal solutions of the problem. Under the additional assumption that u0ââL1(Rn), 0â€u0âLlocp(Rn) for some constant p>n/2, and u0(x)â„Δ/|x|2α for any |x|â„R1 where Δ>0, R1>0, m0<0, α<min(1/(1âm0),1/|m0|), are constants satisfying p>(1âm0)n/2, we prove that the solution of the above problem will converge uniformly on every compact subset of RnĂ(0,â) to the maximal solution of the equation vt=Îlogv, v(x,0)=u0(x), as mâ0â. For any smooth bounded domain ΩâRn, m0<0, mâ[m0,0)âȘ(0,1), and 0â€u0âLp(Ω) for some constant p>(1âm0)max(1,n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem âu/ât=Î(um/m), u>0, in ΩĂ(0,â), u=u0 in Ω, u=g on âΩĂ(0,â) with either finite or infinite positive boundary value g. We also prove a similar convergence result for the solutions of the above Dirichlet problem as mâ0.