On the Cauchy Problem for Hyperbolic Operators with Double Characteristics

2012 ◽  
pp. 247-266 ◽  
Author(s):  
Cesare Parenti ◽  
Alberto Parmeggiani
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Operators ◽  
The Cauchy Problem

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

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