Spectral theory for general nonautonomous/random dispersal evolution operators

Abstract

We investigate the spectral theory of the following general nonautonomous evolution equation ∂ t u ( t , x ) = A ( u ( t , ⋅ ) ) ( x ) + h ( t , x ) u ( t , x ) , x ∈ D , where D is a bounded subset of R N which can be a smooth domain or a discrete set, A is a general linear dispersal operator (for example a Laplacian operator, an integral operator with positive kernel or a cooperative discrete operator) and h ( t , x ) is a smooth function on R × D ¯ . We first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal Lyapunov exponent of a time-dependent dispersal equation is always greater than or equal to that of the time-averaged one. Several results about the principal eigenvalue of time-periodic parabolic equations are extended to general time-periodic dispersal ones. Finally, the investigation is generalized to random time-dependent dispersal equations.