Asymptotic expansion of singular solutions to Protter problem for (2+1)-D degenerate wave equation

Abstract

We consider some boundary value problems for a weakly hyperbolic equation, which are three-dimensional analogues of the Darboux problems on the plain. These problems arise in transonic fluid dynamics and they are introduced by Protter in 1952. As distinct from the planar Darboux problems, the Protter problems are not well posed since the homogeneous adjoint problems have infinitely many nontrivial classical solutions. In 1993 Popivanov and Schneider proved the existence and uniqueness of properly defined generalized solution of a Protter problem and found that this solution may have a strong power-type singularity, which is isolated at one single point and does not propagate along the bicharacteristics. In the present paper we construct asymptotic expansion at the singular point of the generalized solutions for the studied problems. This expansion gives necessary and sufficient conditions on the right-hand side of the equation for existence of singular solutions with fixed order of singularity.