Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by

{ut−γ(t)Δut−Δu=gϵ(t,u), in Ωu=0, on ∂Ω.$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\ \end{array}\right. \end{array}$$

in a smooth bounded domain Ω ⊂ ℝⁿ, n ⩾ 3, where the terms g_ε are a small perturbation, in some sense, of a function f that depends only on u.