Dynamics of Double Time-Delayed Newton–Boussinesq Equations on Unbounded Domains

This paper is concerned with the long term behavior of solutions for Newton–Boussinesq equations with variable delay and distributed delay defined on an unbounded channel. Firstly, the method of uniform tail-estimates is used to surmount the obstacle of the non-compactness of Sobolev embeddings on unbounded domains, and then the existence and uniqueness of tempered pullback attractors is proved. Secondly, we use the idea of Łukaszewicz (Discrete Cont Dyn Syst 34:4211–4222, 2014) to show that the existence of invariant measures. Finally, the upper semicontinuity of pullback attractors as the delay time tends to zero is established.