Abstract :
We present our results in this paper in two parts. In the first part, we consider
the Cauchy problem for the axially symmetric equation x2u y2u kx x u 0
with entire Cauchy data given on an initial planeŽsee Eq.Ž2.1... We solve the
Cauchy problem and obtain its solutions in two cases, depending on whether k is a
positive even integer or k is a positive odd integer. For k odd, we demonstrate that
the solution has more singularities due to the propagation of the singularities of
the coefficients. In the second part, the Cauchy problem for the same equation is
considered, but instead, its entire Cauchy data are given on an initial sphere Žsee
Eq.Ž3.1... Whenever k is a positive even integer, we obtain the global existence of
the solution and determine all possible singularities. Whenever k is a positive odd
integer, we discuss both local and global solutions. As a consequence of our results
in this paper, we show that the Schwarz Potential ConjectureŽsee the Introduction.
for the even dimensional torus is true.
