  The relaxation function theory for a doped two-dimensional Heisenberg
antiferromagnetic system in the paramagnetic state for all wave vectors through
the Brillouin zone is presented in view of low frequency response of high-$T_c$
copper oxide superconductors. We deduced the regions of long lifetime [$T
\lesssim 400(1-4x)$ K] and "overdamped" [$T \gtrsim 700(1-4x)$ K]
paramagnonlike excitations in the temperature ($T$)-doping index ($x$) phase
diagram from plane oxygen nuclear spin-lattice relaxation rate $^{17}(1/T_1)$
data in up to optimally doped La$_{2-x}$Sr$_{x}$CuO$_{4}$ thus providing the
regimes for the spin wave concept and the ''overdamped'' mode.
