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

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(cdot ) to satisfy int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty for all tin 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 fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s})).