Mixed problems or Cauchy problems for semi-degenerate hyperbolic equations of 2-nd order with a parameter (Q1202853)

The author considers mixed problems or Cauchy problems for semi- degenerate hyperbolic equations of second order. She constructs two typical examples of operators: \[ (1)\quad L=\partial^ 2_ t- \rho\partial^ 2_ 1-\partial^ 2_ 2-(\mu+1)\partial_ 1,\qquad(2)\quad L=\partial^ 2_ t-\partial^ 2_ 1-\partial^ 2_ 2-(\mu+1)\rho^{-1}\partial_ 1, \] where \(\mu\) is a parameter, \(\rho\) is a \(C^ \infty\)-function and equals \(x_ 1\) near \(x_ 1=0\). These two examples represent the degenerate case and the singular case, respectively. Then the author discusses existence and uniqueness of solutions of mixed problems or Cauchy problems for these two operators with the paramter ranging in different intervals.