The authors study the singular diffusion equation where Ω⊂ℝn is a bounded domain with appropriately smooth boundary ∂Ω, ρ(x)=dist(x,∂Ω), and prove that if α≥p-1, the equation admits a unique solution subject only to a given initial datum without any boundary value condition, while if 0<α<p-1, for a given initial datum, the equation admits different solutions for different boundary value conditions.