A non-autonomous scalar one-dimensional dissipative parabolic problem: The description of the dynamics

TL;DR

This paper characterizes the structure of non-autonomous attractors for a scalar dissipative parabolic PDE with a time-dependent coefficient, providing a detailed description of the attractor's structure and its gradient properties.

Contribution

It offers a complete description of the pullback attractor structure for the problem, including the construction of non-autonomous equilibria and analysis of limit sets, extending previous results.

Findings

Global attractor has a gradient structure

Complete description of pullback attractor for specific parameter ranges

Construction of non-autonomous equilibria and their connections

Abstract

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem $u_t= u_{xx} + \lambda u - \beta(t)u^3$ when the parameter $\lambda > 0$ varies. Also, we answer a question proposed in [11], concerning the complete description of the structure of the pullback attractor of the problem when $1<\lambda <4$ and, more generally, for $\lambda \neq N^2$, $2 \leq N \in \mathbb{N}$. We construct global bounded solutions , "non-autonomous equilibria", connections between the trivial solution these "non-autonomous equilibria" and characterize the $\alpha$-limit and $\omega$-limit set of global bounded solutions. As a consequence, we show that the global attractor of the associated skew-product flow has a gradient structure. The structure of the related pullback an uniform attractors are derived from that.