Integral representation of a solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument

1997 ◽  
Vol 49 (10) ◽  
pp. 1501-1506
Author(s):  
A. N. Zarubin
Keyword(s):  
Cauchy Problem ◽  
Integral Representation ◽  
Hyperbolic Equation ◽  
Retarded Argument ◽  
The Cauchy Problem

scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

2012 ◽  
Vol 15 (01) ◽  
pp. 1250004 ◽  
Author(s):  
ABDESSATAR BARHOUMI ◽  
HABIB OUERDIANE ◽  
HAFEDH RGUIGUI
Keyword(s):  
Cauchy Problem ◽  
Integral Representation ◽  
White Noise ◽  
Coordinate System ◽  
Functional Integral ◽  
Euler Operator ◽  
The Cauchy Problem ◽  
Sum Formula

In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

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