Asymptotic structure of the attractor for processes on time-dependent spaces

We compare the asymptotic structure of the time-dependent attractor AtAt generated by the partial differential equation εutt+αut−Δu+f(u)=g,εutt+αut−Δu+f(u)=g, where the positive function ε=ε(t)ε=ε(t) tends to zero as t→∞t→∞, with the global attractor A∞A∞ of its formal limit αut−Δu+f(u)=g.αut−Δu+f(u)=g. We establish an abstract result and we apply it to the proof of the convergence At→A∞At→A∞.