  The time evolution of systems relaxing towards thermal equilibrium is
examined near the critical temperature $T_c$, with special attention paid to
the role of the initial value $m_i$ of the order parameter $\phi$. To this end,
the $n$-component model A for a cube of length $L$ is investigated. The common
belief that all memory of $m_i$ is necessarily lost after a microscopic time
span is shown to be unfounded. General arguments and the exact solution of the
limit $n\to\infty$ show that $m_i$ leaves its traces in both the linear and
nonlinear long-time relaxation of $\phi$ near or at $T_c$. Specifically for
linear relaxation near $T_c$, or at $T_c$ with $L<\infty$, the amplitude of the
exponential decay depends on $m_i$ and the short-time exponent
$\theta'=(x_i-x_{\phi})/z$, provided $t_i\sim m_i^{-z/x_i}$ is comparable to or
larger than other time scales. Here $x_i$ is the scaling dimension of $m_i$,
$z$ is the dynamic bulk exponent, and $x_{\phi}$ is the usual equilibrium
scaling dimension of $\phi $.
