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Asymptotics of solutions to the wave equation in a domain with a small hole


Author: D. V. Korikov
Translated by: the author
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 813-838
MSC (2010): Primary 35C20; Secondary 35L05
Published electronically: July 27, 2015
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Abstract: In a bounded domain with a small hole (at all times $ t\in \mathbb{R}$), the wave equation is considered with homogeneous Dirichlet condition on the boundary. The asymptotics of the solution as the diameter of the hole tends to 0 is deduced. To describe the behavior of long waves, the method of compound asymptotic expansions is used. The contribution of short waves (the wavelength is smaller than the diameter of the hole) to the energy of the solution is negligible due to the smoothness of the right-hand side of the wave equation with respect to time.


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Additional Information

D. V. Korikov
Affiliation: Division of Mathematical Physics, Department of Physics, St. Petersburg State University, Ul′yanova 1, St. Petersburg 198504, Russia
Email: thecakeisalie@list.ru

DOI: https://doi.org/10.1090/spmj/1360
Keywords: Hyperbolic equations, singularly perturbed domains, asymptotics of solutions
Received by editor(s): April 7, 2014
Published electronically: July 27, 2015
Additional Notes: Supported by RFBR (grant no. 12-01-00247a) and by SPSU (grant no. 11.38.666.2013)
Article copyright: © Copyright 2015 American Mathematical Society