  The purpose of this paper is to investigate the existence and estimation of
Hausdorff and fractal dimension of global attractors for a delayed
reaction-diffusion equation on an unbounded domain. The noncompactness of the
domain cause the Laplace operator has a continuous spectrum, the semigroup
generated by the linear part and the Sobolev embeddings no longer compact,
making the problem more difficult compared with the equation on bounded domain.
In order to adopt Hausdorff and fractal dimension dimension estimation tools
established for dynamical systems in Hilbert space, we recast the equation in
an auxiliary Hilbert space. We first obtain the existence of global solutions
on unbounded domain by a perturbation semigroup approach which generates an
infinite dimensional dynamical system. Then, we show the existence of global
attractors by firstly showing the existence of an absorbing set and then give a
an uniform a priori estimates for far-field values of solutions which
facilitate us to prove the asymptotic compactness of the generated dynamical
system. After establishing the variational equation in the auxiliary Hilbert
space and the differentiable properties of the generated dynamical system, the
upper estimate of both Hausdorff and fractal dimensions of the global
attractors is obtained.
