On the almost periodicity of nonautonomous evolution equations and application to Lotka-Volterra systems
Abstract
Consider the nonautonomous semilinear evolution equation of type: () \; u'(t)=A(t)u(t)+f(t,u(t)), \; t ∈ R, where A(t), \ t∈ R is a family of closed linear operators in a Banach space X, the nonlinear term f, acting on some real interpolation spaces, is assumed to be almost periodic just in a weak sense (i.e. in Stepanov sense) with respect to t and Lipschitzian in bounded sets with respect to the second variable. We prove the existence and uniqueness of almost periodic solutions in the strong sense (Bohr sense) for equation () using the exponential dichotomy approach. Then, we establish a new composition result of Stepanov almost periodic functions by assuming just the continuity of f in the second variable. Moreover, we provide an application to a nonautonomous system of reaction-diffusion equations describing a Lotka-Volterra predator-prey model with diffusion and time-dependent parameters in a generalized almost periodic environment.