LOCAL EXISTENCE AND UNIQUENESS THEORY FOR THE SECOND SOUND EQUATION IN ONE SPACE DIMENSION

2012 ◽  
Vol 09 (01) ◽  
pp. 177-193 ◽  
Author(s):  
KEIICHI KATO ◽  
YUUSUKE SUGIYAMA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Local Existence ◽  
Second Sound ◽  
Uniqueness Of Solutions ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Uniqueness Theory

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

scholarly journals Some study of a cauchy problem in parabolic integrodifferential equations class

2010 ◽  
Vol 7 (1) ◽  
pp. 101-105
Author(s):  
A. S. AL-FHAID
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Integrodifferential Equations ◽  
Local Existence And Uniqueness ◽  
The Cauchy Problem

We consider the parabolic integrodifferential equations of a form given below. We establish local existence and uniqueness and prove the convergence in L2(En) to the solution Ut of the Cauchy problem.

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.

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.

THE IVP FOR THE SCHRÖDINGER–POISSON-Xα EQUATION IN ONE DIMENSION

2005 ◽  
Vol 15 (08) ◽  
pp. 1169-1180 ◽  
Author(s):  
H. P. STIMMING
Keyword(s):  
Cauchy Problem ◽  
Quantum System ◽  
Existence And Uniqueness ◽  
Space Dimension ◽  
Semiclassical Limit ◽  
Semigroup Theory ◽  
One Dimension ◽  
The Cauchy Problem ◽  
Electron Quantum ◽  
Particle Approximation

The Schrödinger–Poisson-Xα equation is an effective one-particle approximation of a many-electron quantum system. In space dimension d<3, existence analysis for this equation is not contained in standard results for nonlinear Schrödinger equations. We obtain existence and uniqueness of the Cauchy problem in d = 1 using semigroup theory. Furthermore, we discuss the semiclassical limit.

Existence and uniqueness of discontinuous solutions for a hyperbolic system

1997 ◽  
Vol 127 (6) ◽  
pp. 1193-1205 ◽  
Author(s):  
Feimin Huang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Hyperbolic System ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Discontinuous Solutions ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

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