The Cauchy Problem for an Axially Symmetric Equation and the Schwarz Potential Conjecture for the Torus

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.