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November  2020, 19(11): 5181-5196. doi: 10.3934/cpaa.2020232

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

Received  December 2019 Revised  April 2020 Published  September 2020

Fund Project: 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

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.

Citation: Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. M. ArrietaA. 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.  Google Scholar

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Springer, Netherlands, 1976.  Google Scholar

[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. Google Scholar

[7]

H. Brézis, Operateurs Maximaux Monotones, North Holland, Amsterdam, 1973.  Google Scholar

[8]

R. C. D. S. BrocheA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. N. CarvalhoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. ChipotV. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110 (2003), 199-220.   Google Scholar

[16]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, 2002 doi: 10.1007/b83277.  Google Scholar

[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.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Struwe, Variational Methods: Applications in Nonlinear Partial Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. M. ArrietaA. 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.  Google Scholar

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Springer, Netherlands, 1976.  Google Scholar

[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. Google Scholar

[7]

H. Brézis, Operateurs Maximaux Monotones, North Holland, Amsterdam, 1973.  Google Scholar

[8]

R. C. D. S. BrocheA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. N. CarvalhoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. ChipotV. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110 (2003), 199-220.   Google Scholar

[16]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, 2002 doi: 10.1007/b83277.  Google Scholar

[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.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Struwe, Variational Methods: Applications in Nonlinear Partial Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008.  Google Scholar

Figure 1.  Region bounded by the positive equilibria $\phi^+_{1, b_1}$ and $\phi^+_{1, b_2}$
Figure 2.  The set $X^+_2, $ the functions that lie between $\phi^+_{2, b_1}$ and $\phi^+_{2, b_2}$
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