A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation

Abstract

In this paper we study the asymptotic behaviour 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 forward asymptotic behaviour of solutions for models with Kirchhoff type diffusion. In the autonomous case we use the gradient structure, symmetry properties and comparison results to obtain a sequence of bifurcations of equilibria, analogous to what is seen in the local diffusivity case. We provide conditions so that the autonomous problem admits at most one positive equilibrium and analyse the existence of sign changing equilibria. Also using symmetry and the comparison results (developed here) we construct what is called non-autonomous equilibria to describe part of the asymptotics of the associated non-autonomous non-local parabolic problem.

Structure of the attractor for a non-local Chafee-Infante problem

“…In the papers [5] and [30], the authors have studied problem (1) and constructed a sequence of bifurcations similar to the one in the Chafee-Infante equation. To be more precise:…”

“…For more details, see [6] and [30]. In particular, the problems share the same equilibria for each parameter λ > 0.…”

“…In [30], the authors show the existence of a semigroup related to (1) that admits a global attractor A in the phase space H 1 0 (0, π) (see also [6], [5] for other conditions on the nonlinear function f , which include the possibility of non-uniqueness of solutions).…”

“…The asymptotic behavior of solutions as times goes to infinity plays an important role in the study of non-linear partial differential equations. In the last years, several authors have studied the existence and properties of global attractors for such kind of equations in both the autonomous and nonautonomous frameworks (see [2], [7], [11], [10], [9], [30]). Also, non-local equations without uniquenes (see [8], [6]) and with delay [35] have been considered.…”

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

“…For the stationary solutions of (1) we analyze the bifurcations of the equilibria and their hyperbolicity. This analysis has been carried out in [6,2] for the particular case when a is increasing and f is odd (see also [1] for the study of the bifurcation when f is not necessarily odd). These two conditions considerably simplifies the structure of the bifurcations.…”