A mixed problem for some hyperbolic equation with small parameter ε under the presence of a restoring term |u|αu and a reduced problem for a parabolic type are considered. Several ε weighted energy estimates can be obtained by the method of difference quotients. It is shown that the solution of the mixed problem converges, uniformly on any finite time interval, to the solution u of the problem for the parabolic equation in an appropriate Hilbert space as ε→0. © 1996 by the University of Notre Dame. All rights reserved.