Abstract.
We prove that the solution u of the equation u t =Δlog u, u>0, in (Ω\{x 0})×(0,T), Ω⊂ℝ2, has removable singularities at {x 0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C 1, C 2>0, such that C 1 |x−x 0|α≤u(x,t)≤C 2|x−x 0|−α holds for all 0<|x−x 0|≤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≤u 0∉L 1(ℝ2) is radially symmetric and u 0L loc 1(ℝ2).
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Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65
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Hsu, SY. Removable singularities and non-uniqueness of solutions of a singular diffusion equation. Math. Ann. 325, 665–693 (2003). https://doi.org/10.1007/s00208-002-0394-5
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DOI: https://doi.org/10.1007/s00208-002-0394-5