Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation ut=Δ (u) in , where is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at x=0 for t>0, and start with zero initial data. We first prove a nonexistence result when is too sublinear and we show an axial trace result in the other case: there exist a time τ≡τ(u) and a Radon measure λ on [0,τ) such that Then u has a strong singularity on x=0 for any t>τ. We then prove existence of such solutions for any λ and τ as above, and give a uniqueness result for those solutions. Finally, we make a complete study of self-similar solutions (in the power case) which classify the possible asymptotic behaviours.