Forwards attraction properties in scalar non-autonomous linear–dissipative parabolic PDEs. The case of null upper Lyapunov exponent

Abstract

As it is well-known, the forwards and pullback dynamics are in general unrelated. In this paper we present an in-depth study of whether the pullback attractor is also a forwards attractor for the processes involved with the skew-product semiflow induced by a family of scalar non-autonomous reaction-diffusion equations which are linear in a neighbourhood of zero and have null upper Lyapunov exponent. Besides, the notion of Li-Yorke chaotic pullback attractor for a process is introduced, and we prove that this chaotic behaviour might occur for almost all the processes. When the problems are additionally sublinear, more cases of forwards attraction are found, which had not been previously considered even in the case of linear-dissipative ODEs. 2010 Mathematics Subject Classification. 37B55, 35K57, 37L30. Key words and phrases. Non-autonomous dynamical systems; pullback and forwards attraction for processes; linear-dissipative parabolic PDEs; Li-Yorke chaos. R. Obaya and A.M. Sanz were partly supported by FEDER Ministerio de Economía y Competitividad grant MTM2015-66330-P, and the European Commission under project H2020-MSCA-ITN-2014 643073 CRITICS. J.A. Langa was partially supported by Junta de Andalucía under Proyecto de Excelencia FQM-1492 and FEDER Ministerio de Economía y Competitividad grant MTM2015-63723-P.

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow

“…Once more, under assumptions (c1), (c2) and (c3) there exists a global attractor A for τ . This is a consequence of Proposition 3.4 in [17], whose proof works under no further conditions on P rather than compactness.…”

“…for r > 0 and t 0 > 0 both big enough. This is done in Proposition 3.5 in Langa et al [17]. In fact, it can be derived from its proof that the limit in (3.3) exists, so that formula (3.3) is also applicable with Dirichlet boundary conditions.…”

“…All these results remain valid without the minimality condition on P , that is, just assuming that P is compact. The case of Dirichlet boundary conditions has recently been treated in Langa et al [17]. The reader is referred to this paper for all the details.…”

“…We investigate the structure and internal dynamics of the global and cocycle attractors of τ in terms of the principal spectrum of τ L . These questions were initiated in Cardoso et al [3], Caraballo et al [2] and Langa et al [17] when the flow in P is minimal and uniquely ergodic, and the principal spectrum is thus a singleton. Although purely dissipative problems are also considered, these references extensively cover the linear-dissipative case when the flow on P is aperiodic, showing ingredients of high complexity or chaotic dynamics that cannot occur when the equations are autonomous or periodic.…”

On the forward dynamical behavior of nonautonomous systems

“…Unfortunately, there are few results on the existence of such type of attractors except in some particular cases of continuous systems such as the periodic and the asymptotically autonomous ones, and these problems are still undergoing investigations; see e.g. Carvalho et al [8, p. 595], Cheban et al [11], Kloeden et al [29], Langa et al [31], and Wang et al [39]. It is well known that if the pullback attraction of a pullback attractor A = {A(σ)} σ∈Σ of a family of processes {U σ (t, 0)} t 0 , σ ∈ Σ in the space E is uniform with respect to σ ∈ Σ, where Σ is a compact metric space and E is a complete metric space, then it is also a forward attractor.…”