# Cauchy Problem: No Tangent Bicharacteristics

## Abstract

The main purpose of this chapter is to prove two new results on *C*^{ ∞ } well-posedness mentioned in the end of Sect. 1.4. If there is no transition of spectral type and no tangent bicharacteristics then the Cauchy problem is *C*^{ ∞ } well-posed for *P* of order *m* under the strict IPH condition. If the positive trace is zero, the IPH condition is reduced to the Levi condition. In this case we prove that when *p* is of spectral type 2 on *Σ* and there is no tangent bicharacteristics, the Levi condition is necessary and sufficient in order that the Cauchy problem is *C*^{ ∞ } well-posed for *P* of order *m*. The same result holds for second order differential operators of spectral type 1 on *Σ* with 0 positive trace. To prove these assertions using the results in Chap. 2 we first derive microlocal energy estimates. Then making use of the idea developed in Chap. 4 we prove the *C*^{ ∞ } well-posedness of the Cauchy problem.

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