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

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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 ([8]) that the L 2 -norm of the solution itself generally grows to infinity (as t → ∞) in the one and two dimensional cases, however, by adding the potential with quite generous condition 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.