  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.
