The Effect of Domain Squeezing upon the Dynamics of Reaction-Diffusion Equations

AbstractLet Ω be an arbitrary smooth bounded domain in RM×RN and ε>0 be arbitrary. Write (x, y) for a generic point of RM×RN. Squeeze Ω by the factor ε in the y-direction to obtain the squeezed domain Ωε={(x, εy)∣(x, y)∈Ω}. Consider the following reaction-diffusion equation on Ωε:[formula] Here, νε is the exterior normal vector field on ∂Ωε and f:R→R is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that (Eε) generates a semiflow on H1(Ωε) with a global attractor Aε. We prove that, in some strong sense, the equations (Eε) have a limiting equationu+Au=f(u) (E0) as ε→0. This limiting equation is an abstract semilinear parabolic equation which defines a semiflow π on a closed linear subspace of H1(Ω). We show that π has a global attractor A0 and the family of attractors (Aε)ε⩾0 is upper-semicontinuous at ε=0. If M=N=1 and Ω satisfies some natural additional assumptions, then the limiting equation (E0) is equivalent to a parabolic boundary value problem defined on a finite graph. The results of this paper extend previous results obtained by Hale and Raugel for domains which are ordinate sets of a positive function