scholarly journals Global existence of small amplitude solutions to one-dimensional nonlinear Klein–Gordon systems with different masses

2015 ◽  
Vol 12 (04) ◽  
pp. 745-762 ◽  
Author(s):  
Donghyun Kim
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Small Amplitude ◽  
Space Dimension ◽  
Structural Condition ◽  
Cauchy Data ◽  
One Dimensional ◽  
Compactly Supported ◽  
The Cauchy Problem ◽  
Klein Gordon

We study the Cauchy problem for systems of cubic nonlinear Klein–Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the rate [Formula: see text] in [Formula: see text], [Formula: see text] as [Formula: see text] tends to infinity even in the case of mass resonance, if the Cauchy data are sufficiently small, smooth and compactly supported.

scholarly journals The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Hyungjin Huh ◽  
Jihyun Yim
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Space Dimension ◽  
Space Time ◽  
Gauge Condition ◽  
Existence Of Solution ◽  
Temporal Gauge ◽  
The Cauchy Problem ◽  
Temporal And Spatial ◽  
One Space Dimension

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition A1=0 and the temporal gauge condition A0=0.

GLOBAL INHOMOGENEOUS SCHRÖDINGER FLOW

2000 ◽  
Vol 11 (08) ◽  
pp. 1079-1114 ◽  
Author(s):  
HONG-YU WANG ◽  
YOU-DE WANG
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Spin System ◽  
Holomorphic Sectional Curvature ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Schrodinger Flow ◽  
The One ◽  
Complete Kähler Manifold

In this paper, we consider the global existence of one-dimensional nonautonomous inhomogeneous Schrödinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the one-dimensional nonautonomous inhomogeneous Schrödinger flow from S1 into a complete Kähler manifold with constant holomorphic sectional curvature admits a unique global smooth solution. As a corollary, we establish the global existence for the Cauchy problem of the inhomogeneous Heisenberg spin system.

scholarly journals On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jingjing Xu ◽  
Zaihong Jiang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Propagation Speed ◽  
One Dimensional ◽  
Dissipative Term ◽  
The Cauchy Problem ◽  
Infinite Propagation Speed

We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.

scholarly journals On Uniqueness and Continuous Dependence on the Initial Data of the Solution of a System of Two Loaded Parabolic Equations with the Cauchy Data

2019 ◽  
pp. 298-309
Author(s):  
Igor V. Frolenkov ◽  
Irina S. Antipina ◽  
Natalya M. Terskikh
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Parabolic Equations ◽  
Continuous Dependence ◽  
Cauchy Data ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Bounded Functions ◽  
Continuous Dependence Of Solutions

We study the Cauchy problem for the system of one-dimensional loaded parabolic equations. Uniqueness and continuous dependence of solutions on the initial data in the class of smooth bounded functions is proved

scholarly journals Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

2020 ◽  
Vol 23 (6) ◽  
pp. 1663-1677
Author(s):  
Michael Ruzhansky ◽  
Berikbol T. Torebek
Keyword(s):  
Cauchy Problem ◽  
Exponential Function ◽  
Evolution Equations ◽  
Oscillatory Integrals ◽  
Schrödinger Equations ◽  
Fractional Evolution Equations ◽  
Type Function ◽  
Fractional Schrödinger Equations ◽  
The Cauchy Problem ◽  
Klein Gordon

Abstract The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

scholarly journals The cauchy problem for non-linear Klein-Gordon equations

1993 ◽  
Vol 152 (3) ◽  
pp. 433-478 ◽  
Author(s):  
Jacques C. H. Simon ◽  
Erik Taflin
Keyword(s):  
Cauchy Problem ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Klein Gordon
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