  We investigate the energy relaxation process produced by thermal baths at
zero temperature acting on the boundary atoms of chains of classical anharmonic
oscillators. Time-dependent perturbation theory allows us to obtain an explicit
solution of the harmonic problem: even in such a simple system nontrivial
features emerge from the interplay of the different decay rates of Fourier
modes. In particular, a crossover from an exponential to an inverse-square-root
law occurs on a time scale proportional to the system size $N$. A further
crossover back to an exponential law is observed only at much longer times (of
the order $N^3$). In the nonlinear chain, the relaxation process is initially
equivalent to the harmonic case over a wide time span, as illustrated by
simulations of the $\beta$ Fermi-Pasta-Ulam model. The distinctive feature is
that the second crossover is not observed due to the spontaneous appearance of
breathers, i.e. space-localized time-periodic solutions, that keep a finite
residual energy in the lattice. We discuss the mechanism yielding such
solutions and also explain why it crucially depends on the boundary conditions.
