Generalized principal eigenvalues for parabolic operators in bounded domains

TL;DR

This work develops a comprehensive framework of generalized principal eigenvalues for parabolic operators with space-time heterogeneity on bounded domains. It introduces multiple notions (e.g., $\mu_{b,p}$, $\lambda_{b,p}$, $\mu_p$, $\lambda_p$, etc.), relates them to the principal Floquet bundle via the time-global solution $u_P$, and derives fundamental inequalities, perturbation results, and limit-operator characterizations. The authors connect these spectral quantities to the long-time behavior of linear and semilinear (Fisher-KPP type) parabolic equations, provide explicit computations for several coefficient classes (time-invariant, periodic, almost periodic, ergodic), and establish criteria for the maximum principle and the existence/uniqueness of entire solutions. The introduction of the global growth-rate as a unifying tool enables a cohesive treatment of growth dynamics and paves the way for extensions to limit operators and unbounded domains, with potential impact on heterogeneous reaction-diffusion models and Floquet theory generalizations.

Abstract

We introduce here new generalized principal eigenvalues for linear parabolic operators with heterogeneous coefficients in space and time. We consider a bounded spatial domain and an unbounded time interval $I$ : $I=\mathbb{R},\ \mathbb{R}^+$ or $\mathbb{R}^-$, and operators with coefficients having a fairly general dependence on space and time. The notions we introduce rely on the parabolic maximum principle and extend some earlier definitions introduced for elliptic operators [BNV]. We first show that these eigenvalues hold the key to understanding the large time behavior and entire solutions of heterogeneous Fisher-KPP type equations. We then describe the relation of these quantities with principal Floquet bundles for parabolic operators which provides further characterizations of the principal eigenvalues. These allow us to derive monotonicity properties and comparisons between generalized principal eigenvalues, as well as perturbation results and further properties involving limit operators. We show that the sign of these eigenvalues encodes different versions of the maximum principle for parabolic operators. Lastly, we explicitly compute the generalized principal eigenvalues for several classes of operators such as spatial-independent, periodic, almost periodic, uniquely ergodic or random stationary ergodic coefficients.

Theorems & Definitions (55)

• Proposition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 2.1: Huska
• Theorem 2.2: Huska
• Theorem 2.3
• Corollary 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 1