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

Abstract

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