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

TL;DR

This paper investigates the asymptotic behavior of solutions in a non-local, non-autonomous scalar parabolic equation with Kirchhoff-type diffusion, revealing bifurcation structures and equilibrium properties.

Contribution

It provides a comprehensive analysis of bifurcations and equilibria in non-autonomous non-local parabolic equations, extending known results from local diffusion models.

Findings

Bifurcation sequences of equilibria similar to local models.

Conditions for unique positive equilibrium in autonomous case.

Construction of non-autonomous equilibria for asymptotic analysis.

Abstract

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.