Exponential Dichotomies in Higher-Dimensional Spatial Dynamics for Elliptic Partial Differential Equations

TL;DR

The paper extends the spatial dynamics framework to elliptic PDEs on general multi-dimensional domains by proving the existence of far-field exponential dichotomies for the radial evolution. It builds an $r$-dependent norm on $X_r$ to manage the nonuniform angular Laplacian and reduces the principal operator to modewise scalar blocks via spherical harmonics, with a diagonalizable asymptotic potential $V_\infty$ whose eigenvalues drive the dichotomy rate. By establishing an isomorphism $X_r\to L^2\times L^2$ and diagonalizing the system, the authors obtain an ED for the diagonalized problem with rate $\tilde{\eta}=\min_{\ell} \mathrm{Re}[\lambda_\ell^{1/2}]$, then transfer this ED back to the original setting using robustness results. The main theorem provides a constructive tool for analyzing bounded solutions and constructing localized spatial patterns in reaction-diffusion-type problems on general domains, with potential extensions to multi-dimensional Melnikov-type arguments in higher dimensions.

Abstract

Exponential dichotomies, when they exist, provide powerful information about the structure of bounded solutions even in the case of an ill-posed evolutionary equation. The method of spatial dynamics, in which one views a spatial variable as a time-like evolutionary variable, allows for the use of classical dynamical systems techniques, such as exponential dichotomies, in broader contexts. This has been utilized to study stationary, traveling wave, time-periodic, and spiral wave solutions of PDEs on spatial domains with a distinguished unbounded direction (e.g. the real line or a channel of the form $\mathbb{R}\timesΩ$). Recent work has shown how to extend the spatial dynamics framework to elliptic PDEs posed on general multi-dimensional spatial domains. In this paper, we show that, in the same context, exponential dichotomies exist, thus allowing for their use in future analyses of coherent structures, such as spatial patterns in reaction-diffusion equations on more general domains.

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 2.1
• proof : Proof (of Lemma \ref{['l.isomorphism']}).
• proof : Proof (of Lemma \ref{['l.ED_diagonalized']}).
• proof : Proof (of Theorem \ref{['thrm.ED_existence']}).
• Lemma A.1