  In this paper we study the asymptotic behavior of solutions for a non-local
non-autonomous scalar quasilinear parabolic problem in one space dimension. Our
aim is to give a fairly complete description of the the forwards asymptotic
behavior of solutions for models with Kirchoff type diffusion. In the
autonomous we use the gradient structure of the model, some symmetry properties
of solutions and develop comparison results to obtain a sequence of
bifurcations of equilibria analogous to that seen in the model with local
diffusivity. We give conditions so that the autonomous problem admits at most
one positive equilibrium and analyse the existence of sign changing equilibria.
Also using symmetry and our comparison results we construct what is called
non-autonomous equilibria to describe part of the asymptotics of the associated
non-autonomous non-local parabolic problem.
