scholarly journals Singularities of Solutions to the Cauchy Problem for a Class of Second-Order Hyperbolic Operators

2014 ◽  
Vol 57 (3) ◽  
pp. 375-448
Author(s):  
Seiichiro Wakabayashi
Keyword(s):  
Cauchy Problem ◽  
Second Order ◽  
Hyperbolic Operators ◽  
The Cauchy Problem

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 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.

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

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