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Existence of singular solutions of a degenerate equation in R 2

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Abstract

Let a 1,a 2, . . . ,a m ∈ ℝ2, 2≤f ∈ C([0,∞)), g i ∈ C([0,∞)) be such that 0≤g i (t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation u t =Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−a i |→−g i (t) as |x−a i |→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u 0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain.

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  1. Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi, 621, Taiwan

    Shu-Yu Hsu

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  1. Shu-Yu Hsu
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Hsu, SY. Existence of singular solutions of a degenerate equation in R 2 . Math. Ann. 334, 153–197 (2006). https://doi.org/10.1007/s00208-005-0714-7

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  • DOI: https://doi.org/10.1007/s00208-005-0714-7

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