Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations

Abstract

We consider the Dirichlet problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz domains. Using a new Harnack-type inequality for quotients of positive solutions, we show that each positive solution exponentially dominates any solution which changes sign for all times. We then examine continuity and robustness properties of a principal Floquet bundle and the associated exponential separation under perturbations of the coefficients and the spatial domain. RésuméOn considère le problème de Dirichlet pour des équations paraboliques linéaires non autonomes du second ordre avec coefficients bornés mesurables sur un domaine borné de Lipschitz. Utilisant une nouvelle inégalité du type Harnack pour les quotients de solutions strictement positives, on montre que toute solution positive domine exponentiellement toute solution qui change de signe en tout temps. On examine ensuite les propriétés de continuité et de robustesse pour un fibré principal de Floquet et la séparation exponentielle associée à des perturbations des coefficients et du domaine spatial.

Localized Non-diffusive Asymptotic Patterns for Nonlinear Parabolic Equations with Gradient Absorption

“…In fact, the area of asymptotic behaviour of parabolic equations with variable coefficients is quite open even in the linear case. We refer to [24] for an interesting recent development for linear parabolic equations.…”

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: divergence case

“…There are many other robustness results that one can address in connection with exponential separations, see [16] for an account of such results for bounded Lipschitz domains. Many of these results can be carried over to the problem on R N using the methods of the present paper.…”

Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$

“…Indeed, as we demonstrate in the last section, our methods are straightforward to adapt to the more general setting. If R N in (1.1) is replaced by Ω, a bounded domain in R N , and (1.1) is complemented with a suitable boundary condition, there is a number of papers devoted to properties of solutions of (1.1) analogous to properties of principal eigenfunctions of time-independent (elliptic) or time-periodic parabolic problems, see for example [11,12,14,15,16,24,25,26,29,32,33]. Typical results can briefly be summarized as follows.…”

“…The exponential separation is a generalization of the fact that the principal eigenvalue of an elliptic operator is larger than the real part of any other eigenvalue. See [14,16,25,26] for a more detailed discussion of the connection of principal Floquet bundles with principal eigenvalues and eigenfunctions. The study of principal Floquet bundles and exponential separation in nonautonomous parabolic equations with general time-dependence originated in [21], [33].…”