Principal Eigenvalues of Generalized Convolution Operators on the Circle and Spreading Speeds of Noncompact Evolution Systems in Periodic Media

Source abstract

The main aim of this paper is to understand what kind of diffusion mechanism can guarantee the existence of spreading speeds for evolution systems in periodic media. The work in this paper is presented in three parts. First, the uniform irreducibility of finite Radon measures on the circle is defined, and it is proved that a generalized convolution operator generated by such a Radon measure admits a principal eigenvalue. Next, an abstract framework of the existence and characterization of spreading speeds for general spatially periodic noncompact systems is established, under the hypothesis that the linearized systems have principal eigenvalues. Finally, based on the above preparation, it is shown that the uniform irreducibility of diffusion can guarantee the existence of spreading speeds in periodic media by investigating an integro-difference equation and a nonlocal diffusion equation with Kolmogorov--Petrovskii--Piskunov nonlinearity.