  We consider a nonlinear reaction-diffusion equation settled on the whole
euclidean space. We prove the well-posedness of the corresponding Cauchy
problem in a general functional setting, namely, when the initial datum is
uniformly locally bounded in L^2. Then we adapt the short trajectory method to
establish the existence of the global attractor and, if the space dimension is
at most 3, we also find an upper bound of its Kolmogorov's entropy.
