Nonlocal and Initial Problems for Quasilinear, Nonstrictly Hyperbolic Equations with General Solutions Represented by Superposition of Arbitrary Functions

2003 ◽  
Vol 10 (4) ◽  
pp. 687-707
Author(s):  
J. Gvazava
Keyword(s):  
Cauchy Problem ◽  
General Solution ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Quasilinear Equations ◽  
General Solutions ◽  
Parabolic Degeneracy

Abstract We have selected a class of hyperbolic quasilinear equations of second order, admitting parabolic degeneracy by the following criterion: they have a general solution represented by superposition of two arbitrary functions. For equations of this class we consider the initial Cauchy problem and nonlocal characteristic problems for which sufficient conditions are established for the solution solvability and uniquness; the domains of solution definition are described.

scholarly journals Find A General Solution Of An Equation Of The Hyperbolic Type With A Second-Order Singular Coefficient And Solve The Cauchy Problem Posed For This Equation.

2021 ◽  
Vol 25 (1) ◽  
pp. 80
Author(s):  
Karimova Shalola Musayevna ◽  
Melikuzieva Dilshoda Mukhtorjon qizi
Keyword(s):  
Cauchy Problem ◽  
General Solution ◽  
Type Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Singular Coefficient ◽  
The Cauchy Problem

This paper presents a general solution of a hyperbolic type equation with a second-order singular coefficient and a solution to the Cauchy problem posed for this equation.

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

scholarly journals On the Cauchy problem for finitely degenerate hyperbolic equations of second order

2000 ◽  
Vol 38 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Ferruccio Colombini ◽  
Haruhisa Ishida ◽  
Nicola Orrú
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equations ◽  
Second Order ◽  
The Cauchy Problem ◽  
Degenerate Hyperbolic Equations

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.

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