The Cauchy Problem, Boundary Value and Mixed Problems for Hyperbolic systems in the Complete Scale of Sobolev Type Spaces

1992 ◽  
pp. 168-175
Author(s):  
Ja. A. Roitberg
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Sobolev Type ◽  
Mixed Problems ◽  
The Cauchy Problem ◽  
Sobolev Type Spaces ◽  
Type Spaces

scholarly journals Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

scholarly journals Approximate solutions of equations defined by complex spherical multiplier operators

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-115
Author(s):  
C. P. Oliveira
Keyword(s):  
Approximate Solutions ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Sobolev Type ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Solutions Of Equations ◽  
Multiplier Operators

This paper studies, in a partial but concise manner, approximate solutions of equations defined by complex spherical multiplier operators. The approximations are from native spaces embedded in Sobolev-type spaces and derived from the use of positive definite functions to perform spherical interpolation.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

scholarly journals Boundary value problems with nonlocal conditions for hyperbolic systems of equations with two independent variables

2020 ◽  
Vol 53 (2) ◽  
pp. 159-180
Author(s):  
V. M. Kyrylych ◽  
O. Z. Slyusarchuk
Keyword(s):  
Boundary Value Problems ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Problems ◽  
Priori Estimates ◽  
Smooth Coefficients

Nonlocal boundary value problems for arbitrary order hyperbolic systems with one spatial variable are considered. A priori estimates for general nonlocal mixed problems for systems with smooth and piecewise smooth coefficients are obtained. The correct solvability of such problems is proved.Examples of additional conditions necessity are provided.

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