On the Existence and Uniqueness of Solutions of the CAUCHY Problem for Linear Partial Differential – functional Equations of the First Order

1977 ◽  
Vol 80 (1) ◽  
pp. 183-200 ◽  
Author(s):  
Z. Kamont
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Uniqueness Of Solutions ◽  
Partial Differential ◽  
First Order ◽  
The Cauchy Problem

Sufficient conditions of a nonlocal solvability for a system of two quasilinear equations of the first order with constant terms

2020 ◽  
Vol 55 ◽  
pp. 60-78
Author(s):  
M.V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

We consider a Cauchy problem for a system of two quasilinear equations of the first order with constant terms. The study of the solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms in the original coordinates is based on the method of an additional argument. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. We prove the existence and uniqueness of the local solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms, which has the same smoothness with respect to $x$ as the initial functions of the Cauchy problem. Sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms are found; this solution is continued by a finite number of steps from the local solution. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms relies on global estimates.

scholarly journals Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rui Li ◽  
Xing Lin ◽  
Zongwei Ma ◽  
Jingjun Zhang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Energy Conservation ◽  
Classical Solution ◽  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Zakharov System ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system.

Some Results for the Davey-Stewartson System on a Circle

2012 ◽  
Vol 204-208 ◽  
pp. 4429-4432
Author(s):  
Tsai Jung Chen ◽  
Yung Fu Fang ◽  
Ying Ji Hong
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
System Model ◽  
Mathematical Aspect ◽  
Uniqueness Of Solutions ◽  
Apriori Estimates ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem of the Davey-Stewartson system on a circle. We establish, from a mathematical aspect, certain apriori estimates necessary to ensure the existence and uniqueness of solutions of the Davey-Stewartson system model.

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