A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations

Abstract

In this paper, we consider the asymptotic behavior of traveling wave solutions of a certain degenerate nonlinear parabolic equation for ξ ≡ x − ct → −∞ with c > 0. We give a refined one of them, which was not obtained in the preceding work [Ichida-Sakamoto, J. Elliptic and Parabolic Equations, to appear], by an appropriate asymptotic study and properties of the Lambert W function.

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

“…Here, the meaning of the classification of this solutions comes from the fact that we have revealed all dynamics to infinity of the ordinary differential equations (hereinafter, ODEs) obtained by introducing traveling wave coordinates by using these methods. In addition, Ichida-Matsue-Sakamoto [12] gave a refined asymptotic behavior, which was not obtained in the preceding work [15], by an appropriate asymptotic study and properties of the Lambert W function.…”

“…The conclusions about the traveling wave solution in (1), obtained without requiring p ∈ N, can be reflected in (2) by u = U p . This means that we can obtain a generalized results without the restriction p ∈ N in the conclusion obtained for (2) ( [15,12]). This automatically proves the existence of unproven connecting orbits in the case that p is odd, which was an issue in these studies under considering nonnegative solutions.…”

“…See Fig. 1 in [12] for schematic picture of this solution. Note that ( 12) is consistent with the result obtained in Theorem 2 of [12].…”

“…In addition, this equation with δ = 0 is such an equation that has many phenomena in its background. See, for instance, [4,15,12,23] and references therein for background phenomena. For instance, in the curve shortening problem, we can see that the classical curvature flow equation V = −κ corresponds to (2) in the case that p = 2 (see [5,9,10]).…”

“…With the motivation of contributing to the derivation of a more refined blowup rate by examining in more detail the traveling wave solution studied by [23], in [15,12], they restrict to 1 < p ∈ N, set µ = 1, and give the classification of traveling wave solutions of (2) for both δ = 0 and δ = 1. Ichida-Sakamoto [15] shows that results on the classification (existence only for p ∈ 2N, shapes, and asymptotic behavior) of (weak) traveling wave solutions are given by applying the dynamical system approach, the Poincaré compactification, and geometric methods for desingularization of vector fields called blow-up technique (for instance, see [8] for the details of the Poincaré compactification and blow-up technique).…”