Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

Goro Akagi, Petri Juutinen, Ryuji Kajikiya

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17 Citations (Scopus)

Abstract

The asymptotic behavior of viscosity solutions to the Cauchy-Dirichlet problem for the degenerate parabolic equation u t = Δ u in Ω × (0,∞), where Δ stands for the so-called infinity-Laplacian, is studied in three cases: (i) Ω = R Nand the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of Δφ = 0.

Original languageEnglish
Pages (from-to)921-953
Number of pages33
JournalMathematische Annalen
Volume343
Issue number4
DOIs
Publication statusPublished - 2009 Apr
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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