Continuity of attractors for net-shaped thin domains - Elsken, Thomas (Topological Methods in Nonlinear Analysis, 2005)
Consider a reaction-diffusion equation u_t=u+f(u) on a family of net-shaped thin domains Ω_ε converging to a one dimensional set as ε0. With suitable growth and dissipativeness conditions on f these equations define global semiflows which have attractors A_ε. In [Th. Elsken, < i> A reaction-diffusion equation on a net-shaped thin domain< /i> , Studia Math. < b> 165< /b> (2004), 159–199] it has been shown that there is a limit problem which also defines a semiflow having an attractor A_0, and the family of attractors is upper-semi-continuous at ε=0. Here we show that under a stronger dissipativeness condition the family of attractors A_ε, ε0, is actually continuous at ε=0.