Well-Posedness of the Cauchy Problem for Some Degenerate Hyperbolic Operators

2006 ◽  
pp. 23-41
Author(s):  
Alessia Ascanelli ◽  
Massimo Cicognani
Keyword(s):  
Cauchy Problem ◽  
Well Posedness ◽  
Hyperbolic Operators ◽  
The Cauchy Problem

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.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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