Dynamics of non-autonomous nonclassical diffusion equations on $R^n$

We consider the Cauchy problem for a non-autonomous nonclassical diffusion equation of the form $u_t-\varepsilon\Delta u_t - \Delta u+f(u)+\lambda u=g(t)$ on $R^n$. Under an arbitrary polynomial growth order of the nonlinearity $f$ and a suitable exponent growth of the external force $g$, using the method of tail-estimates and the asymptotic a priori estimate method, we prove the existence of an $(H^{1}(R^n) L^{p}(R^n), H^{1}(R^n) L^{p}(R^n))$ - pullback attractor $\hat{A}_{\varepsilon}$ for the process associated to the problem. We also prove the upper semicontinuity of $\{\hat{A}_{\varepsilon}: \varepsilon\in [0,1]\}$ at $\varepsilon = 0$.