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

Abstract

Decay estimates in the supremum norm for the solutions to a nonlinear evolution equation Abstract. We study the asymptotic behavior, as t → ∞, of the solutions to the nonlinear evolution equationis the normalized p-Laplace equation and p ≥ 2. We show that if u(x, t) is a viscosity solution to the above equation in a cylinder Ω × (0, ∞) with time-independent lateral boundary values, then it converges to the unique stationary solution h as t → ∞. Moreover, we provide an estimate for the decay rate of max x∈Ω |u(x, t) − h(x)|.

Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian

“…In [PR16] the authors obtained local Hölder and Lipschitz estimates using a game theoretic method for the case p = p(x, t) > 2. The asymptotic behavior for (1.1) has been investigated in [Juu14,BG13,Doe11].…”

Hölder gradient estimates for a class of singular or degenerate parabolic equations

“…This is the most relevant case for applications to tug-of-war-like stochastic games with white noise, see Peres-Sheffield [20]. This equation has been studied by Garofalo [9], Banerjee-Garofalo [1,2,3], Does [8], Manfredi-Parviainen-Rossi [17,18], Rossi [21], Juutinen [13], Kawohl-Krömer-Kurtz [14], Liu-Schikorra [16], Rudd [22], as well as the last two authors [12]. Hölder estimates for the spatial gradient of their solutions was proved in [12].…”

Overdetermined problems for the normalized 𝑝-Laplacian

“…The normalized p-Laplacian has also been studied in the context of evolution equations in a number of papers, see [11,8,4,3,20,12,10].…”