Resumen
Let be an arbitrary smooth bounded domain in M×N and >0 be arbitrary. Write (x, y) for a generic point of M×N. Squeeze by the factor in the y-direction to obtain the squeezed domain ={(x, y) | (x, y)}. Consider the following reaction-diffusion equation on : Here, is the exterior normal vector field on and f: is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that (E) generates a semiflow on H1() with a global attractor . We prove that, in some strong sense, the equations (E) have a limiting equation as 0. This limiting equation is an abstract semilinear parabolic equation which defines a semiflow on a closed linear subspace of H1(). We show that has a global attractor 0 and the family of attractors ()0 is upper-semicontinuous at =0. If M=N=1 and 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.