On the Energy Estimate for Klein–Gordon-Type Equations with Time-Dependent Singular Mass

Hirosawa, Fumihiko

doi:10.1007/978-3-030-04459-6_31

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Proof Of the Theorem1

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2022

2023

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Cited by 2 publications

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“…The proof of the theorem is based on the methods introduced in [4, 5,6] that the Klein-Gordon type equation is reduced to a dissipative wave equation or a wave equation with time dependent propagation speed. Then, solutions of the equations are estimated in a particular zones of time-frequency space by the method introduced in [7,8] after the Fourier transformation with respect to spatial variables.…”

Section: Proof Of the Theoremmentioning

confidence: 99%

On the wave-like energy estimates of Klein-Gordon type equations with time dependent potential

Goto¹,

Hirosawa²

2022

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We consider the conditions for the time dependent potential in which the energy of the Cauchy problem of Klein-Gordon type equation asymptotically behaves like the energy of the wave equation. The conclusion of this paper is that the condition is not always given by the order of the potential itself, but should be given by "generalized zero mean condition", which is represented by the integral of the potential. We also introduce "generalized modified energy conservation" in order to describe the appropriate energy for our problem.

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Section: Proof Of the Theoremmentioning

confidence: 99%