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

Let Ω be an arbitrary smooth bounded domain in M× N and >0 be arbitrary. Write (x, y) for a generic point of M× N. Squeeze Ω by the factor in the y-direction to obtain the squeezed domain Ω ={(x, y) (x, y) Ω}. Consider the following reaction-diffusion equation on Ω :[formula] Here, ν is the exterior normal vector field on ∂Ω and f: → is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that (E ) generates a semiflow on H1(Ω ) with a global attractor . We prove that, in some strong sense, the equations (E ) have a limiting equationu+Au=f(u) (E0) as →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 0 and the family of attractors ( ) 0 is upper-semicontinuous at =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.