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

✍ Scribed by Hong Liu

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 140 KB
• Volume: 250
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.6964

✦ Synopsis

We present our results in this paper in two parts. In the first part, we consider k 2 2

the Cauchy problem for the axially symmetric equation Ѩ u q Ѩ u q Ѩ u s 0

. with entire Cauchy data given on an initial plane see Eq. 2.1 . 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 Ž considered, but instead, its entire Cauchy data are given on an initial sphere see Ž .. Eq. 3.1 . 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 Ž . in this paper, we show that the Schwarz Potential Conjecture see the Introduction for the even dimensional torus is true.