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

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) and 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 . Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455), and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology. Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology.