  The purpose of this paper is to investigate the existence and Hausdorff
dimension as well as fractal dimension of global attractors for a delayed
reaction-diffusion equation on an unbounded domain. The noncompactness of the
domain causes the Laplace operator has a continuous spectrum, the semigroup
generated by the linear part and the Sobolev embeddings are no longer compact,
making the problem more difficult compared with the equations on bounded
domains. We first obtain the existence of an absorbing set for the infinite
dimensional dynamical system generated by the equation by a priori estimate of
the solutions. Then, we show the asymptotic compactness of the solution
semiflow by an uniform a priori estimates for far-field values of solutions
together with the Arzel\`a-Ascoli theorem, which facilitates us to show the
existence of global attractors. By decomposing the solution into three parts
and establishing a squeezing property of each part, we obtain the explicit
upper estimation of both Hausdorff and fractal dimension of the global
attractors, which only depend on the inner characteristic of the equation,
while not related to the entropy number compared with the existing literature.
