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

摘要

We study the Cauchy problem for the inhomogeneous two-dimensionalwave equation with variable coe cients and zero initial data. The righthandside is assumed to be localized in space and time. The equation isconsidered in a domain with a boundary (shore). The velocity is assumedto 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 theboundary of the domain where the problem is studied. The main result ofthe paper is e cient asymptotic formulas for the solution of this problem,including the neighborhood of the shore.