Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains

The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor u(t+θ) in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes.