  In this papwe we consider an effective role of the potential of the wave
equations with/without damping on the L^{2}-estimate of the solution itself. In
the free wave equation case it is known that the L^{2}-norm of the solution
itself generally grows to infinity (as time goes to infinity) in the one and
two dimensional cases, however, by adding the potential with quite generous
conditions one can controle the growth property to get the L^{2}-bounds. This
idea can be also applied to the damped wave equations with potential in order
to get fast energy and L^{2} decay results in the low dimensional case, which
are open for a long period. Applications to heat and plate equations with a
potential can be also studied. In this paper the low dimensional case is a main
target.
