Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation

Carvalho, Alexandre N.; Langa, José A.; Robinson, James C.

doi:10.1090/s0002-9939-2011-11071-2

Cited by 23 publications

(35 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, g might just be the map given by g(p, x, y) =    k(p, x) (y + r 0 ) 3 , y ≤ −r 0 0 , −r 0 ≤ y ≤ r 0 −k(p, x) (y − r 0 ) 3 , y ≥ r 0 for a certain positive map k ∈ C(P ×Ū ) and the constant r 0 in (c4), which provides a non-autonomous version of the classical Chafee-Infante equation. The autonomous equation was studied by Chafee and Infante [10] and some non-autonomous versions of this equation together with bifurcation problems have also been treated in the literature; for instance, see Carvalho et al [8].…”

Section: 2mentioning

confidence: 99%

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

Caraballo

Langa

Obaya

et al. 2018

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g. in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.

show abstract

Section: 2mentioning

confidence: 99%

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

Caraballo

Langa

Obaya

et al. 2018

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…3 Partially supported by FEDER and Ministerio de Economía y Competitividad # grant MTM2012-30860 and by Junta de Castilla y León under project VA118A12-1. Carvalho et al [Carvalho et al (2012)] study the asymptotic behaviour of the following non-autonomous version of the Chafee-Infante equation:…”

Section: Introductionmentioning

confidence: 99%

Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

Cardoso

Langa

Obaya

2016

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

show abstract

“…Moreover, we also know precisely the diagram of connections between equilibria (see [17]) and that this diagram is stable under autonomous and nonautonomous perturbations (see [18,2,6]). In addition, when we replace b in (1.2) by a time dependent function which is not close to a constant, there has been interesting developments ensuring that the asymptotics still resembles that of (1.2) with b constant (see [11,8]). The introduction of a non-local diffusion changes everything.…”

Section: (12)mentioning

confidence: 99%

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

Li¹,

Carvalho²,

Luna³

et al. 2020

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

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.

show abstract

Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation

Cited by 23 publications

References 18 publications

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

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

Contact Info

Product

Resources

About