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

TL;DR

This paper investigates conditions under which pullback attractors in scalar non-autonomous linear dissipative PDEs also serve as forward attractors, especially when the upper Lyapunov exponent is zero, revealing new cases of forward attraction.

Contribution

It provides a detailed analysis of forward attraction properties in scalar non-autonomous PDEs with null Lyapunov exponent, including new cases in sublinear problems.

Findings

Pullback attractors can also be forward attractors under certain conditions.

Chaotic behavior can occur for almost all processes in the studied class.

New forward attraction cases are identified for sublinear problems.

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.