Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations

A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows

“…In fact, µ(λ) is related to the almost sure principal Lyapunov exponent of a heat operator with random potential [38], known as the parabolic Anderson problem ( [10,12] and references). Dynamical aspects of principal Lyapunov exponents as an extension of principal eigenvalues are recently studied in [24]. Regularity of µ(λ) is an interesting problem in itself.…”

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

“…This research was motivated by some open questions in nonlinear parabolic equations, in particular, the problem of typical asymptotic behavior in periodic-parabolic equations (see [32,33,9]). Other applications can be found in [15,16,17,22,23,24,25,26,37]. We also mention that in one space dimension Floquet bundles corresponding to any nodal number can be established, thus extending the classical Sturm-Liouville theorem (see [4,5,38]).…”

“…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].…”

Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations

“…The reader is also referred to [9][10][11]13,[17][18][19][20][21]25] for the studies of principal spectrum of various nonautonomous parabolic equations. Roughly speaking, the principal spectrum of the problem (c; B) is the principal eigenvalue of the associated eigenvalue problem of (1.2).…”