Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes

1977 ◽  
Vol 17 (6) ◽  
pp. 921-931 ◽  
Author(s):  
V. Ya. Ivrii
Keyword(s):  
Cauchy Problem ◽  
Gevrey Classes ◽  
Variable Multiplicity ◽  
Hyperbolic Operators

scholarly journals Inhomogeneous Gevrey ultradistributions and Cauchy problem

2006 ◽  
Vol 133 (31) ◽  
pp. 176-186
Author(s):  
Daniela Calvo ◽  
L. Rodino
Keyword(s):  
Cauchy Problem ◽  
Weight Function ◽  
Differential Operators ◽  
Subject Classification ◽  
Well Posedness ◽  
Gevrey Classes ◽  
Partial Differential Operators ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Constant Coefficients

After a short survey on Gevrey functions and ultradistributions, we present the inhomogeneous Gevrey ultradistributions introduced recently by the authors in collaboration with A. Morando, cf. [7]. Their definition depends on a given weight function ?, satisfying suitable hypotheses, according to Liess-Rodino [16]. As an application, we define (s, ?)-hyperbolic partial differential operators with constant coefficients (for s > 1), and prove for them the well-posedness of the Cauchy problem in the frame of the corresponding inhomogeneous ultradistributions. This sets in the dual spaces a similar result of Calvo [4] in the inhomogeneous Gevrey classes, that in turn extends a previous result of Larsson [14] for weakly hyperbolic operators in standard homogeneous Gevrey classes. AMS Mathematics Subject Classification (2000): 46F05, 35E15, 35S05.

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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