The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in literature, we do not ask the symbol space to have finite box-counting dimension. Instead, we ask a condition on the semi-continuity of pullback attractors of the system as time goes to infinity. This semi-continuity can be achieved if we suppose the existence of finite-dimensional exponential uniform attractors for the limit symbols. After showing these new results, we apply them to study the box-counting dimension of the uniform attractor for a reaction-diffusion equation, and we find a specific forcing term such that the symbol space has infinite box-counting dimension but the uniform attractor has finite box-counting dimension anyway.
