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

A role of potential on L^{2}-estimates for some evolution equations