Abstract
We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples.
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Supported by Grants-in-Aid for Young Scientists (B) (No.16740098), The Ministry of Education, Culture, Sports, Science and Technology.
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Hirosawa, F. On the asymptotic behavior of the energy for the wave equations with time depending coefficients. Math. Ann. 339, 819–838 (2007). https://doi.org/10.1007/s00208-007-0132-0
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DOI: https://doi.org/10.1007/s00208-007-0132-0