scholarly journals On the Cauchy problem for a class of hyperbolic operators whose coefficients depend only on the time variable

2015 ◽  
Vol 39 (1) ◽  
pp. 121-163 ◽  
Author(s):  
Seiichiro Wakabayashi
Keyword(s):  
Cauchy Problem ◽  
Time Variable ◽  
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.

scholarly journals Fractional-hyperbolic systems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Fractional Derivative ◽  
Hyperbolic Systems ◽  
Time Variable ◽  
Spatial Variables ◽  
First Order ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Evolution Systems

AbstractWe describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order α ∈ (0, 1) in the time variable t and the first order derivatives in spatial variables x = (x 1, …, x n), which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone {(t,x) : |t -α| ≤ 1}.

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