Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(\cdot )$ to satisfy $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{L^{2}}\,ds<\infty $ for all $t\in \mathbb{R}$ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $\mathscr{D}_{\lambda}$ -attractor indeed can attract in the $H^{1}$ -norm, provided that $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{H^{-1}(\mathcal{O}_{s})}\,ds< \infty $ and $f\in L^{2}_{\mathrm{loc}}(\mathbb{R},L^{2}(\mathcal{O}_{s}))$ .