Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions

1999 ◽  
Vol 313 (1) ◽  
pp. 127-140 ◽  
Author(s):  
Tohru Ozawa ◽  
Kimitoshi Tsutaya ◽  
Yoshio Tsutsumi
Keyword(s):  
Cauchy Problem ◽  
Energy Space ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
Three Space

scholarly journals ON THE WAVE EQUATION WITH QUADRATIC NONLINEARITIES IN THREE SPACE DIMENSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
AXEL GRÜNROCK
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Parameter Range ◽  
Well Posedness ◽  
Quadratic Nonlinearities ◽  
The Cauchy Problem ◽  
Three Space

The Cauchy problem for the nonlinear wave equation [Formula: see text] in three space dimensions is considered. The data (u0, u1) are assumed to belong to [Formula: see text], where [Formula: see text] is defined by the norm [Formula: see text] Local well-posedness is shown in the parameter range 2 ≥ r > 1, [Formula: see text]. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal on the Hs-scale by Lindblad's counterexamples, but nonetheless leaves a gap of ½ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for □u = ∂u2 are obtained, too.

scholarly journals Klein-Gordon Equations on Modulation Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Guoping Zhao ◽  
Jiecheng Chen ◽  
Weichao Guo
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Stability Theory ◽  
Scattering Theory ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Blowup Criterion ◽  
The Cauchy Problem ◽  
Klein Gordon

We consider the Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spacesMp,1a. We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

scholarly journals THE CAUCHY PROBLEM FOR METRIC-AFFINE f(R)-GRAVITY IN PRESENCE OF A KLEIN–GORDON SCALAR FIELD

2011 ◽  
Vol 08 (01) ◽  
pp. 167-176 ◽  
Author(s):  
S. CAPOZZIELLO ◽  
S. VIGNOLO
Keyword(s):  
Cauchy Problem ◽  
Scalar Field ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Klein Gordon

We study the initial value formulation of metric-affine f(R)-gravity in presence of a Klein–Gordon scalar field acting as source of the field equations. Sufficient conditions for the well-posedness of the Cauchy problem are formulated. This result completes the analysis of the same problem already considered for other sources.

scholarly journals Global well-posedness in energy space for the Chern–Simons–Higgs system in temporal gauge

2016 ◽  
Vol 13 (02) ◽  
pp. 331-351 ◽  
Author(s):  
Hartmut Pecher
Keyword(s):  
Cauchy Problem ◽  
Energy Space ◽  
Well Posedness ◽  
Null Condition ◽  
Temporal Gauge ◽  
Time Estimates ◽  
Dimensional Minkowski Space ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

The Cauchy problem for the Chern–Simons–Higgs system in the [Formula: see text]-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d’Ancona, Foschi and Selberg, an [Formula: see text]-estimate for solutions of the wave equation, and also takes advantage of a null condition.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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