Removable singularities and non-uniqueness of solutions of a singular diffusion equation

Abstract

We prove that the solution u of the equation ut=Δlog u, u>0, in (Ω\{x0})×(0,T), Ω⊂ℝ2, has removable singularities at {x0}×(0,T) if and only if for any 0 0, such that C1|x−x0|α≤u(x,t)≤C2|x−x0|−α holds for all 0<|x−x0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u0∉L1(ℝ2) is radially symmetric and u0Lloc1(ℝ2).