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The time evolution of systems relaxing towards thermal equilibrium isexamined near the critical temperature $T_c$, with special attention paid tothe 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 commonbelief that all memory of $m_i$ is necessarily lost after a microscopic timespan is shown to be unfounded. General arguments and the exact solution of thelimit $n\to\infty$ show that $m_i$ leaves its traces in both the linear andnonlinear long-time relaxation of $\phi$ near or at $T_c$. Specifically forlinear relaxation near $T_c$, or at $T_c$ with $L<\infty$, the amplitude of theexponential 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 orlarger 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 equilibriumscaling dimension of $\phi $.Comment: 12 pages, uuencoded postscript file (yesterday an old version of this  manuscript was submitted by mistake!!

Long-time traces of the initial condition in relaxation phenomena near criticality