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

2023/08/30 by Motohiro Sobajima, Sobajima, Motohiro
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.15680

openalex publication_date 2023/08/30 · openalex created_date 2023/09/02 · openalex updated_date 2026/07/28

Abstract

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 Schrödinger 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.

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