Non-autonomous reaction–diffusion model with dynamic boundary conditions

We investigate the dynamics of a non-autonomous reaction–diffusion model with dynamic boundary conditions. We first show that, under the same assumptions, the known L2(Ω)×L2(∂Ω)L2(Ω)×L2(∂Ω) pullback DD-attractor indeed can attract in L2+δ(Ω)×L2+δ(∂Ω)L2+δ(Ω)×L2+δ(∂Ω)-norm for any δ∈[0,∞)δ∈[0,∞); then we prove the continuity of the solution in H1(Ω)×H12(∂Ω) with respect to the initial data, and finally show that such attractor can also attract in H1(Ω)×H12(∂Ω)-norm under a slightly stronger integrability condition on the time-dependent external forcing term. The proofs are based on a new Nash–Moser–Alikakos type a priori estimate about the difference of solutions near the initial time.