Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time

Anatoly Anikin

Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow,119526, Russia Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow
Region, 141701, Russia
E-mail: anikin83@inbox.ru

Sergey Dobrokhotov

Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow,119526, Russia Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow
Region, 141701, Russia
E-mail: nazaikinskii@yandex.ru

Vladimir Nazaikinskii

Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow,119526, Russia Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow
Region, 141701, Russia
E-mail: nazaikinskii@yandex.ru

Received March 19, 2018.

Dedicated to the 95th anniversary of V.A. Marchenko, a prominent scientist and great personality

Abstract

We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The righthand
side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore.