  We study the non-equilibrium dynamics (purely dissipative and relaxational)
in a semi-infinite system following a quench from the high temperature
disordered phase to its critical temperature. We show that the local
autocorrelation near the surface of a semi-infinite system decays algebraically
in time with a new exponent which is different from the bulk. We calculate this
new non-equilibrium surface exponent in several cases, both analytically and
numerically.
