On explicit formulas for a~solution to the Darboux problem and the~Cauchy--Goursat problem for a~degenerate hyperbolic equation (Q1307152)

The author considers the hyperbolic equation \[ y^mu_{xx}-u_{yy}=0, \quad m>0, \tag{1} \] in the domain bounded by a segment of hyperbolic degeneration \(AB\) and the characteristics variable families \(AC\) and \(BC\). The author also writes down explicit formulas for solutions to four boundary value problems in the case when the Dirichlet data is given on one of the characteristics \(AC\) or \(BC\) and the Dirichlet or Neumann boundary condition is given on \(AB\). The formulas by the author are simpler than those available. In addition, the author studies the properties of solutions at the corner points \(A\) and \(B\) and gives more precise sufficient conditions (as compared with those available) for the classical solutions to the equation to be continuous up to the boundary.