scholarly journals Solution of the cauchy problem for a hyperbolic equation with constant coefficients in the case of two independent variables

2012 ◽  
Vol 48 (5) ◽  
pp. 707-716 ◽  
Author(s):  
V. I. Korzyuk ◽  
I. S. Kozlovskaya
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Constant Coefficients ◽  
Two Independent Variables

scholarly journals On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

2020 ◽  
Vol 17 (01) ◽  
pp. 75-122
Author(s):  
Ferruccio Colombini ◽  
Tatsuo Nishitani
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Second Order ◽  
Gevrey Class ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Two Independent Variables

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

Uniqueness In Boundary Value Problems For The Second Order Hyperbolic Equation

1956 ◽  
Vol 8 ◽  
pp. 86-96 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Second Order ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Surface ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Order Hyperbolic Equation ◽  
Second Order Hyperbolic Equation ◽  
The Uniqueness Theorem

Introduction. We study linear normal hyperbolic partial differential equations of the second order, with one dependent variable u, and N independent variables xi (i = 1, … , N). The uniqueness theorem connected with the Cauchy problem for this type of equation is well known and in effect states that if u and its first normal derivatives vanish on a spacelike initial surface S then u vanishes in a certain conical region which contains S (1, p. 379).

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