A reaction-diffusion equation on a net-shaped thin domain
Thomas Elsken · Studia Mathematica · 2004
Let Ω ε ⊂ R M +1 , 0 < ε ≤ 1, be a net-shaped Lipschitz domain which collapses to a one-dimensional net as ε ↓ 0. On Ω ε we consider the equation u t = ∆u with von Neumann boundary conditions.We show under quite general conditions that the semiflows generated by this equation have a limit in a strong sense, the limit semiflow being generated by an abstract linear operator.Also, under an additional assumption, the eigenvalues and eigenfunctions of the corresponding operators converge.This allows us to apply the techniques in [14] to prove the convergence of the nonlinear semiflows generated by a reaction-diffusion equation on Ω ε and the upper-semicontinuity of their attractors at ε = 0. Our technique also allows us to treat the case that Ω ε is smooth and has holes which vanish of order at least ε in all directions.