  We prove spatiotemporal algebraically decaying estimates for the density of
the solutions of the linearly damped nonlinear Schr\"odinger equation with
localized driving, when supplemented with vanishing boundary conditions. Their
derivation is made via a scheme, which incorporates suitable weighted Sobolev
spaces and a time-weighted energy method. Numerical simulations examining the
dynamics (in the presence of physically relevant examples of driver types and
driving amplitude/linear loss regimes), showcase that the suggested decaying
rates, are proved relevant in describing the transient dynamics of the
solutions, prior their decay: they support the emergence of waveforms
possessing an algebraic space-time localization (reminiscent of the Peregrine
soliton) as first events of the dynamics, but also effectively capture the
space-time asymptotics of the numerical solutions.
