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

Yanan Li; Alexandre N. Carvalho; Tito L. M. Luna; Estefani M. Moreira

doi:10.3934/cpaa.2020232

Article Contents

2020, Volume 19, Issue 11: 5181-5196. Doi: 10.3934/cpaa.2020232

This issue Previous Article Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations Next Article On the inelastic Boltzmann equation for soft potentials with diffusion

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

1.
College of Mathematical Sciences, Harbin Engineering University, 150001, China
2.
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil

^* Corresponding author
^* Corresponding author

Received: December 2019

Revised: April 2020

Early access: September 2020

Published: November 2020

The first author was supported by NSFC Grant 11671367. The second author was supported by Grants FAPESP 2018/10997-6 and CNPq 306213/2019-2. The third author was supported by FAPESP Grant 2019/20341-3. The fourth author was supported by Grants FAPESP 2018/00065-9 and CAPES-Scholarship 7547361/D

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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.

Keywords:

One dimensional scalar parabolic problem,
non-local nonlinear diffusion,
comparison principles,
bifurcation,
non-autonomous equilibria.

Mathematics Subject Classification: Primary: 35B09, 35B51, 35B32; Secondary: 35B41, 35B06, 35B40.

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Figure 1. Region bounded by the positive equilibria $\phi^+_{1, b_1}$ and $\phi^+_{1, b_2}$

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Figure 2. The set $X^+_2, $ the functions that lie between $\phi^+_{2, b_1}$ and $\phi^+_{2, b_2}$

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