Abstract: Based on the process theory in time-dependent space, we considered the long-time dynamic behavior of the solution for the non-damping suspension bridge equation with fading memory. Firstly, we obtained well-posedness of solution by using Faedo-Galerkin approximation method. Secondly, the non-autonomous dynamical system had pullback absorb set in the corresponding solution space was obtained by using energy estimation. Finally,  we proved the existence of time-dependent pullback attractors by using  the  contraction function method and cocyclic technique.

Key words: suspension bridge equation, pullback absorb set, time-dependent pullback attractor, fading memory