Spectral theory for general nonautonomous/random dispersal evolution operators

Abstract

We investigate the spectral theory of the following general nonautonomous evolution equation

$${\partial }_{t}u(t,x)=\mathcal{A}(u(t,\cdot ))(x)+h(t,x)u(t,x),x\in D,$$

where D is a bounded subset of ${\mathbb{R}}^N$ which can be a smooth domain or a discrete set, $\mathcal{A}$ 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 $\mathbb{R}\times \overline{D}$. 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.