A non-autonomous Chafee–Infante equation

Abstract

In this chapter we study the existence and characterisation of pullback attractors for a non-autonomous version of the Chafee–Infante equation on the domain (0, π),

$${u}_{t} - {u}_{xx} = \lambda u - b(t){u}^{3},\qquad u(0,t) = u(\pi ,t) = 0,$$

(13.1)

when there exist 0 < b ₀ < B ₀ such that

$$0 < {b}_{0} \leq b(t) \leq {B}_{0}.$$

Theorem 12.1 guarantees the local existence and uniqueness of solutions for an initial u(s) ∈ H ₀ ¹(0, π), and Proposition 12.8 ensures that these solutions are globally defined. Monotonicity properties of the solutions of this equation (from Corollary 12.6) will play an essential role in our analysis, and we will be able to prove the existence of maximal and minimal bounded global solutions, ξ _m ( ⋅) and ξ _M ( ⋅), which provide ‘bounds’ on the asymptotic dynamics of the system, i.e. any bounded global solution ψ( ⋅) satisfies

$${\xi }_{m}(t) \leq \psi (t) \leq {\xi }_{M}(t)\qquad \mbox{ for all}\qquad t \in \mathbb{R}.$$