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: November 30, 2019

Revised: March 31, 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

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

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.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	C. O. Alves and F. J. S. A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Commun. Appl. Nonlinear Anal., 8 (2001), 43-56.
[2]	S. B. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differ. Equ., 62 (1986), 427-442. doi: 10.1016/0022-0396(86)90093-8.
[3]	S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.
[4]	J. M. Arrieta, A. N. Carvalho and A. Rodriguez-Bernal, Attractors of parabolic problems with nonlinear boundary condition. Uniform bounds, Commun. Partial Differ. Equ., 25 (2000), 1-37. doi: 10.1080/03605300008821506.
[5]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Springer, Netherlands, 1976.
[6]	M. C. Bortolan, A. N. Carvalho, J. A. Langa and G. Raugel, Non-autonomous perturbations of Morse-Smale semigroups: stability of the phase diagram, preprint.
[7]	H. Brézis, Operateurs Maximaux Monotones, North Holland, Amsterdam, 1973.
[8]	R. C. D. S. Broche, A. N. Carvalho and J. Valero, A non-autonomous scalar one-dimensional dissipative parabolic problem: the description of the dynamics, Nonlinearity, 32 (2019), 4912-4941. doi: 10.1088/1361-6544/ab3f55.
[9]	A. N. Carvalho and C. B. Gentile, Comparison results for nonlinear parabolic equations with monotone principal part, J. Math. Anal. Appl., 259 (2001), 319-337. doi: 10.1006/jmaa.2001.7506.
[10]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
[11]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation, Proc. Amer. Math. Soc., 140 (2012), 2357-2373. doi: 10.1090/S0002-9939-2011-11071-2.
[12]	N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974), 17–37. doi: 10.1080/00036817408839081.
[13]	N. Chafee and E. F. Infante, Bifurcation and stability for a nonlinear parabolic partial differential equation, Bull. Amer. Math. Soc., 80 (1974), 49-52. doi: 10.1090/S0002-9904-1974-13349-5.
[14]	X. Y. Chen and H. Matano, Convergence, asymptotic periodicity and finite-point blow-up in one dimensional semilinear heat equations, J. Differ. Equ., 78 (1989), 160-190. doi: 10.1016/0022-0396(89)90081-8.
[15]	M. Chipot, V. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110 (2003), 199-220.
[16]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, 2002 doi: 10.1007/b83277.
[17]	B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differ. Equ., 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031.
[18]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.
[19]	H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), 401–441.
[20]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[21]	A. Rodriguez-Bernal and A. Vidal-Lopez, Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems, Discrete Contin. Dyn. Syst., 18 (2007), 537-567. doi: 10.3934/dcds.2007.18.537.
[22]	M. Struwe, Variational Methods: Applications in Nonlinear Partial Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008.