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Mat. Zametki, 2017, Volume 101, Issue 5, Pages 700–715 (Mi mz11489)  

Uniform Asymptotics of the Boundary Values of the Solution in a Linear Problem on the Run-Up of Waves on a Shallow Beach

S. Yu. Dobrokhotovab, V. E. Nazaikinskiiab, A. A. Tolchennikovab

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

Abstract: We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain $\Omega$. The velocity is assumed to degenerate on the boundary $\partial\Omega$ of the domain as the square root of the distance to $\partial\Omega$. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associated with the problem can be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

Keywords: wave equation, nonstandard characteristics, run-up on a shallow beach, localized source, asymptotics, boundary values, trace theorem, higher-order transport equations.

Funding Agency Grant Number
Russian Science Foundation 16-11-10282

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm11489

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English version:
Mathematical Notes, 2017, 101:5, 802–814

Bibliographic databases:

UDC: 517.9
Received: 02.12.2016

Citation: S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, “Uniform Asymptotics of the Boundary Values of the Solution in a Linear Problem on the Run-Up of Waves on a Shallow Beach”, Mat. Zametki, 101:5 (2017), 700–715; Math. Notes, 101:5 (2017), 802–814

Citation in format AMSBIB
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\paper Uniform Asymptotics of the Boundary Values of the Solution in a~Linear Problem on the Run-Up of Waves on a~Shallow Beach
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\vol 101
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\pages 700--715
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