The Cauchy problem for the system of Zakharov equations arising from ion-acoustic modes

In this paper the initial value problem for a class of Zakharov equations arising from ion-acoustic modes is discussed. Without assuming the Cauchy data are small, we prove the existence and uniqueness of the global smooth solution for the problem via the so-called continuous method and delicate a priori estimates.

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Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1990.0009 ◽

1990 ◽

Vol 427 (1872) ◽

pp. 221-239 ◽

Cited By ~ 111

Keyword(s):

Cauchy Problem ◽

Perfect Fluid ◽

Initial Value Problem ◽

Einstein Equations ◽

Initial Value ◽

Fluid Source ◽

Null Hypersurfaces ◽

The Cauchy Problem ◽

Well Posed ◽

Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

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Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

Discrete Dynamics in Nature and Society ◽

10.1155/2016/4021516 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8

Author(s):

Minoru Tabata ◽

Nobuoki Eshima

Keyword(s):

Cauchy Problem ◽

Initial Value Problem ◽

Equilibrium Solution ◽

Global Solutions ◽

Initial Time ◽

Spatial Economics ◽

Replicator Equation ◽

Unique Global Solution ◽

Initial Value ◽

The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

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Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽

10.3390/universe5120224 ◽

2019 ◽

Vol 5 (12) ◽

pp. 224 ◽

Cited By ~ 1

Author(s):

Stefano Vignolo

Keyword(s):

Cauchy Problem ◽

Initial Value Problem ◽

Sufficient Conditions ◽

Well Posedness ◽

Junction Conditions ◽

Field Equations ◽

Initial Value ◽

The Cauchy Problem ◽

General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

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On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

Abstract and Applied Analysis ◽

10.1155/2012/535031 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Hatice Taskesen ◽

Necat Polat ◽

Abdulkadir Ertaş

Keyword(s):

Cauchy Problem ◽

Initial Value Problem ◽

Type Equation ◽

Global Solutions ◽

Initial Energy ◽

Potential Well ◽

Global Weak Solutions ◽

Power Type ◽

Initial Value ◽

The Cauchy Problem

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

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On the convergence of the method of finite differences applied to the solution of the initial value problem for functional-differential neutral type equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500020278 ◽

1981 ◽

Vol 89 (3-4) ◽

pp. 259-265

Author(s):

N. G. Kazakova ◽

D. D. Bainov

Keyword(s):

Cauchy Problem ◽

Initial Value Problem ◽

Finite Differences ◽

Neutral Type ◽

Integral Type ◽

Initial Value ◽

Natural Class ◽

Quadrature Formulae ◽

The Cauchy Problem ◽

Functional Differential

SynopsisWhen solving practically the neutral type equations the derivatives are replaced by finite differences while the members of integral type are replaced by quadrature formulae. The paper deals with the convergence of a natural class of methods applied to the Cauchy problem for functional-differential neutral type equations. It is not obligatory for the approximated operators to be compact.

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An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018062 ◽

2019 ◽

Vol 14 (2) ◽

pp. 204

Author(s):

Eduardo Hernandez-Montero ◽

Andres Fraguela-Collar ◽

Jacques Henry

Keyword(s):

Inverse Problem ◽

Cauchy Problem ◽

Laplace Equation ◽

Boundary Data ◽

A Priori ◽

Normal Derivative ◽

Cauchy Data ◽

Dirichlet Data ◽

Data Completion ◽

The Cauchy Problem

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

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Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Abstract and Applied Analysis ◽

10.1155/2013/931493 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Josef Diblík ◽

Michal Fečkan ◽

Michal Pospíšil

Keyword(s):

Cauchy Problem ◽

Differential Equations ◽

Initial Value Problem ◽

Linear Differential Equations ◽

Multiple Delays ◽

Initial Value ◽

Matrix Polynomials ◽

The Cauchy Problem ◽

Nonhomogeneous System ◽

Linear Differential

Nonhomogeneous system of linear differential equations of second order with multiple different delays and pairwise permutable matrices defining the linear parts is considered. Solution of corresponding initial value problem is represented using matrix polynomials.

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Estimates for Solutions of Wave Equations with Vanishing Curvature

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-064-9 ◽

1985 ◽

Vol 37 (6) ◽

pp. 1176-1200 ◽

Cited By ~ 2

Author(s):

Bernard Marshall

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Cauchy Problem ◽

Linear Combination ◽

Initial Value Problem ◽

Wave Equations ◽

Hyperbolic Partial Differential Equation ◽

Initial Value ◽

The Cauchy Problem ◽

Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

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ON THE CAUCHY PROBLEM IN BESOV SPACES FOR A NON-LINEAR SCHRÖDINGER EQUATION

Communications in Contemporary Mathematics ◽

10.1142/s0219199700000116 ◽

2000 ◽

Vol 02 (02) ◽

pp. 243-254 ◽

Cited By ~ 15

Author(s):

FABRICE PLANCHON

Keyword(s):

Cauchy Problem ◽

Schrödinger Equation ◽

Initial Value Problem ◽

Schrodinger Equation ◽

Initial Value ◽

Non Linear ◽

The Cauchy Problem ◽

Self Similar ◽

Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011308 ◽

1980 ◽

Vol 21 (1) ◽

pp. 65-80 ◽

Cited By ~ 3

Author(s):

S. Nababan ◽

K.L. Teo

Keyword(s):

Cauchy Problem ◽

Weak Solution ◽

Differential Equations ◽

Delay Differential Equations ◽

Existence And Uniqueness ◽

A Priori ◽

Divergence Form ◽

Partial Delay ◽

The Cauchy Problem ◽

Delay Differential

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

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