Pullback attractors for nonclassical diffusion equations with a delay operator

The paper proves existence of a unique pullback D-attractor in CH_t(Ω) and its boundedness in CH^1_t(Ω) for the nonclassical diffusion equation with delay (1.1), via weighted energy estimates, a tempered absorbing set, a v1+v2 decomposition, fractional smoothing (σ<min{1/3,(n+2−(n−2)γ)/2}), and ω-limit compactness. The candidate solution reaches the same conclusions using a closely related, but not identical, route: a Lyapunov functional with an exponentially weighted history term, direct fractional testing with A^σ, Simon-type compactness, and a single-level Au test for regularity on the attractor. Assumptions and key constants (e.g., Ca1>3/2+L/2+1/(4λ1), β1>0) match the paper’s hypotheses, and the logical steps are sound. Differences are methodological (decomposition vs. direct Simon compactness; a different but equivalent way of handling the delay term), not substantive.