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Stability of traveling waves for doubly nonlinear equations

Christian Seis, Dominik Winkler

TL;DR

The paper proves stability and high-regularity for the pressure of solutions to the doubly nonlinear diffusion equation in the slow diffusion regime, near traveling-wave profiles. By translating the moving boundary problem into traveling-wave coordinates and employing a graphed variable $\zeta$, the authors derive a perturbation equation $\partial_t w + \mathcal{L}_\sigma w = \mathcal{N}[w]$ with a degenerate linear operator $\mathcal{L}_\sigma$ and an analytic nonlinear term $\mathcal{N}[w]$. They develop an abstract fixed-point/implicit-function framework in tailored Banach spaces to obtain existence, uniqueness, and analyticity in time and tangential directions, and they verify the assumptions for the perturbation equation using maximal regularity results (Kienzler) and Angenent's trick. This yields quantitative derivative bounds and analyticity of level sets for the pressure near the traveling-wave front, extending prior results for the porous medium equation to the broader doubly nonlinear diffusion setting. The work provides a rigorous foundation for the stability of free-boundary behavior and smoothness propagation in slow-diffusion flows with nonlinear diffusion operators.

Abstract

In this note, we investigate a doubly nonlinear diffusion equation in the slow diffusion regime. We prove stability of the pressure of solutions that are close to traveling wave solutions in a homogeneous Lipschitz sense. We derive regularity estimates for arbitrary derivatives of the solution's pressure by extending existing results for the porous medium equation (Ref. 15).

Stability of traveling waves for doubly nonlinear equations

TL;DR

The paper proves stability and high-regularity for the pressure of solutions to the doubly nonlinear diffusion equation in the slow diffusion regime, near traveling-wave profiles. By translating the moving boundary problem into traveling-wave coordinates and employing a graphed variable , the authors derive a perturbation equation with a degenerate linear operator and an analytic nonlinear term . They develop an abstract fixed-point/implicit-function framework in tailored Banach spaces to obtain existence, uniqueness, and analyticity in time and tangential directions, and they verify the assumptions for the perturbation equation using maximal regularity results (Kienzler) and Angenent's trick. This yields quantitative derivative bounds and analyticity of level sets for the pressure near the traveling-wave front, extending prior results for the porous medium equation to the broader doubly nonlinear diffusion setting. The work provides a rigorous foundation for the stability of free-boundary behavior and smoothness propagation in slow-diffusion flows with nonlinear diffusion operators.

Abstract

In this note, we investigate a doubly nonlinear diffusion equation in the slow diffusion regime. We prove stability of the pressure of solutions that are close to traveling wave solutions in a homogeneous Lipschitz sense. We derive regularity estimates for arbitrary derivatives of the solution's pressure by extending existing results for the porous medium equation (Ref. 15).
Paper Structure (3 sections, 7 theorems, 62 equations)

This paper contains 3 sections, 7 theorems, 62 equations.

Table of Contents

  1. Introduction and Main Theorem
  2. New coordinates and perturbation equation
  3. Derivation of Theorem \ref{['maintheorem']}

Key Result

Theorem 1.1

There exists an $\varepsilon>0$ such that for any nonnegative $\rho_0$ with $\rho_0^{\frac{m+p-3}{p-1}}\in \dot{C}^{0,1}\left(\mathbb{R}^n\right)$ satisfying there exists a solution $\rho$ of the doubly nonlinear diffusion equation 1 with initial datum $\rho_0$ satisfying for a constant $c>0$ and additionally for any $k \in \mathbb{N}_0$, $\beta\in \mathbb{N}_0^n$ and a constant $\tilde{c} = \t

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition A: Kienzler16
  • Lemma 2.2
  • Theorem 2.3