Pullback permanence for non-autonomous partial differential equations

A system of differential equations is permanent if there exists a fixed bounded set of positive states strictly bounded away from zero to which, from a time on, any positive initial data enter and remain. However, this fact does not happen for a differential equation with general non-autonomous terms. In this work we introduce the concept of pullback permanence, defined as the existence of a time dependent set of positive states to which all solutions enter and remain for suitable initial time. We show this behaviour in the non-autonomous logistic equation u(t) - Delta u = lambda u - b(t)u(3), with b(t) > 0 for all t is an element of R, lim(t ->infinity) b(t) = 0. Moreover, a bifurcation scenario for the asymptotic behaviour of the equation is described in a neighbourhood of the first eigenvalue of the Laplacian. We claim that pullback permanence can be a suitable tool for the study of the asymptotic dynamics for general non-autonomous partial differential equations.