Time averaging for nonautonomous/random linear parabolic equations

TL;DR

This paper investigates the spectral properties of nonautonomous and random linear parabolic PDEs, establishing relationships between principal eigenvalues of time-averaged equations and spectral exponents, with implications for stability analysis.

Contribution

It extends the theory of principal spectrum and Lyapunov exponents to nonautonomous and random parabolic equations with time-dependent boundary conditions.

Findings

Principal eigenvalue of time-averaged equation is not larger than the supremum of the principal spectrum.

Principal eigenvalue of time-averaged equation is not larger than the principal Lyapunov exponent in the random case.

Existence of a time-averaged equation with principal eigenvalue not larger than the infimum of the principal spectrum.

Abstract

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.