Persistence properties of solutions of dynamical systems in Banach spaces

In this paper, we establish general results for the asymptotic behaviour of solutions of dynamical systems in Banach spaces. We show that if the initial datum possesses a certain decay, then the corresponding solution emanating from the Cauchy problem studied inherits the same behaviour at any further time for which it exists. Our results are applied to a wide class of linear and non-linear models. In particular, we use our main results to show persistence properties for classical linear and non-linear ordinary differential equations (ODEs), the Benjamin–Bona–Mahony (BBM) equation, and the generalized Boussinesq equation.