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

Abstract

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

Keywords

• Semilinear heat equation
• Time-varying domain
• Pullback attractor
• Continuity