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(· ) to satisfy ∫ _-∞^te^λ sf(s)^2_L^2 ds<∞ for all t∈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 𝒟_λ -attractor indeed can attract in the H^1 -norm, provided that ∫ _-∞^te^λ sf(s)^2_H^-1(𝒪_s) ds< ∞ and f∈ L^2_loc(ℝ,L^2(𝒪_s)) .