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<p>It is known that the asymptotic behavior of time-dependent dissipative coefficient in the Cauchy problem of dissipative wave equation dominates the energy decay estimate. In particular, it is important to study the case where the dissipative coefficient behaves like 1/<i>t</i> as <i>t</i> goes to infinity, which is called weak dissipation, because its order is close to the critical case of decay and non-decay. In this case, an oscillating perturbation of weak dissipation can give a crucial effect on the energy decay estimate, but the analysis is very difficult compared to the case without oscillations. In this paper, we develop a method recently introduced by Ghisi-Gobbino that has contributed to a precise analysis for dissipative wave equations with oscillating weak dissipation, and consider the effect of the oscillations, which is more general and close to the critical case. Furthermore, we study the effect of the smoothness of the initial data on the energy decay estimates.</p>

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On energy decay estimates for weak dissipative wave equations with oscillating time-dependent coefficient in the dissipative term

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • Fumihiko Hirosawa,
  • Daichi Nakajima

摘要

It is known that the asymptotic behavior of time-dependent dissipative coefficient in the Cauchy problem of dissipative wave equation dominates the energy decay estimate. In particular, it is important to study the case where the dissipative coefficient behaves like 1/t as t goes to infinity, which is called weak dissipation, because its order is close to the critical case of decay and non-decay. In this case, an oscillating perturbation of weak dissipation can give a crucial effect on the energy decay estimate, but the analysis is very difficult compared to the case without oscillations. In this paper, we develop a method recently introduced by Ghisi-Gobbino that has contributed to a precise analysis for dissipative wave equations with oscillating weak dissipation, and consider the effect of the oscillations, which is more general and close to the critical case. Furthermore, we study the effect of the smoothness of the initial data on the energy decay estimates.