2020/07/02 by Khalil, Kamal
#34G10 #47D06 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2007.01143
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.