Attractor of the nonclassical diffusion equation with memory on time- dependent space

We consider the dynamic behavior of solutions for a nonclassical diffusion equation with memory

$$u_t-\epsilon (t)\mathrm{△}{u}_t-\mathrm{△}u-{\int }_0^{\mathrm{\infty }}\kappa (s)\mathrm{△}u(t-s)ds+f(u)=g(x)$$

on time-dependent space for which the norm of the space depends on the time $t$ explicitly, and the nonlinear term satisfies the critical growth condition. First, based on the classical Faedo-Galerkin method, we obtain the well-posedness of the solution for the equation. Then, by using the contractive function method and establishing some delicate estimates along the trajectory of the solutions on the time-dependent space, we prove the existence of the time-dependent global attractor for the problem. Due to very general assumptions on memory kernel $\kappa$ and the effect of time-dependent coefficient $\epsilon (t)$, our result will include and generalize the existing results of such equations with constant coefficients. It is worth noting that the nonlinear term cannot be treated by the common decomposition techniques, and this paper overcomes the difficulty by dealing with it as a whole.