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

Abstract

The asymptotic behavior of viscosity solutions to the Cauchy-Dirichlet problem for the degenerate parabolic equation u t = ∆ ∞ u in Ω × (0, ∞), where ∆ ∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) Ω = R N 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 ∆ ∞ φ = 0.

Asymptotics of viscosity solutions to some doubly nonlinear parabolic equations

“…g(x, t), ∀(x, t) ∈ ∂Ω × [0, T ) 1 We take h ∈ C(P T ), in the sense that lim y→x f (y) = g(x, 0) for each x ∈ ∂Ω. In most of this work, we take…”

“…From the works in [1,10] one sees that (i) for ∆ ∞ u = u t , the asymptotic decay is t −1/2 and (ii) for ∆ p u = u t , the rate is t −1/(p−2) . They do appear to agree if we consider G p .…”

On the viscosity solutions to a degenerate parabolic differential equation

“…They show that solutions are Lipschitz continuous. In [2], the authors study asymptotic behavior of solutions to the same equation. The work in [12] presents a characterization of sub-solutions.…”

Decay estimates in the supremum norm for the solutions to a nonlinear evolution equation

“…By arguing as in [1], one can verify that u(x, t) → h(x) exponentially fast also if |Dh(x)| ≥ c > 0 in Ω for some c > 0. We conjecture that the exponential decay is true in general, that is, the decay estimate in Theorem 3.1 can be improved to u(x, t) − h(x) L ∞ (Ω) ≤ Ce −λt for some λ, C > 0.…”

“…Since we are not aware of a method that would enable us to approximate h by solutions with non-vanishing gradient (unless p = ∞, see [1]), we instead replace h by a solution h ε of an approximating equation −∆ N p h ε = ε and set v(x, t) = h ε (x) + f (t, h ε (x)). Then the above computation yields…”