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

TL;DR

This paper investigates the dynamics of non-autonomous scalar parabolic PDEs over a compact flow, distinguishing between linear-dissipative and purely dissipative cases, and analyzes their attractors using spectral theory.

Contribution

It provides a detailed analysis of the structure of attractors for these PDEs, introducing conditions to identify nontrivial sections in the purely dissipative case.

Findings

Structure of global and pullback attractors characterized

Conditions for nontrivial attractor sections established

Differences between linear-dissipative and purely dissipative cases clarified

Abstract

In this paper a family of non-autonomous scalar parabolic PDEs over a general compact and connected flow is considered. The existence or not of a neighbourhood of zero where the problems are linear has an influence on the methods used and on the dynamics of the induced skew-product semiflow. That is why two cases are distinguished: linear-dissipative and purely dissipative problems. In both cases, the structure of the global and pullback attractors is studied using principal spectral theory. Besides, in the purely dissipative setting, a simple condition is given, involving both the underlying linear dynamics and some properties of the nonlinear term, to determine the nontrivial sections of the attractor.