Optimal decay for one-dimensional damped wave equations with potentials via a variant of Nash inequality

The optimality of decay properties of the one-dimensional damped wave equations with potentials belonging to a certain class is discussed. The typical ingredient is a variant of Nash inequality which involves an invariant measure for the corresponding Schr\"odinger semigroup. This enables us to find a sharp decay estimate from above. Moreover, the use of a test function method with the Nash-type inequality provides the decay estimate from below. The diffusion phenomena for the damped wave equations with potentials are also considered.