Non-autonomous reaction-diffusion equations with variable exponents and large diffusion

In this work we prove continuity of solutions with respect to initial conditions and a couple of parameters and we prove upper semicontinuity of a family of pullback attractors for the problem

$\left\{ {\begin{array}{*{20}{l}}{\frac{{\partial {u_s}}}{{\partial t}}(t) - {D_s}{\rm{div}}(|\nabla {u_s}{|^{{p_s}(x) - 2}}\nabla {u_s}) + C(t)|{u_s}{|^{{p_s}(x) - 2}}{u_s} = B({u_s}(t)),\;\;t > \tau ,}\\{{u_s}(\tau ) = {u_{\tau s}},}\end{array}} \right.$

under homogeneous Neumann boundary conditions, $u_{τ s}∈ H: = L^2(Ω),$ $Ω\subset\mathbb{R}^n$($n≥ 1$) is a smooth bounded domain, $B:H \to H$ is a globally Lipschitz map with Lipschitz constant $L≥ 0$, $D_s∈[1,∞)$, $C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$ is bounded from above and below and is monotonically nonincreasing in time, $p_s(·)∈ C(\bar{Ω})$, $p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$ $p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$ for all $s∈ \mathbb{N},$ when $p_s(·) \to p$ in $L^∞(Ω)$ and $D_s \to ∞$ as $s \to∞,$ with $a,p>2$ positive constants.