scholarly journals Estimates for Unimodular Multipliers on Modulation Hardy Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan ◽  
Lijing Sun ◽  
Chunjie Zhang
Keyword(s):  
Cauchy Problem ◽  
Hardy Spaces ◽  
Asymptotic Estimate ◽  
Nonlinear Partial Differential Equations ◽  
Modulation Spaces ◽  
Mixed Norm ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Norm Estimates ◽  
The Cauchy Problem

It is known that the unimodular Fourier multiplierseit|Δ|α/2,α>0,are bounded on all modulation spacesMp,qsfor1≤p,q≤∞. We extend such boundedness to the case of all0<p,q≤∞and obtain its asymptotic estimate astgoes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study theMp1,qs→Mp2,qsboundedness for the operatoreit|Δ|α/2, for the case0<α≤2andα≠1.Finally, we investigate the boundedness of the operatoreit|Δ|α/2forα>0and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroupeit|Δ|α/2.

scholarly journals Dilation Properties for Weighted Modulation Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Elena Cordero ◽  
Kasso A. Okoudjou
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
Sharp Estimate ◽  
Modulation Spaces ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Unweighted Case ◽  
Vibrating Plate

We give a sharp estimate on the norm of the scaling operatorUλf(x)=f(λx)acting on the weighted modulation spacesMs,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

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 Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

2021 ◽  
Vol 56 (2) ◽  
pp. 176-184
Author(s):  
S.M. Bak ◽  
G. M. Kovtonyuk
Keyword(s):  
Cauchy Problem ◽  
Infinite System ◽  
Global Solutions ◽  
Well Posedness ◽  
Operator Differential Equation ◽  
Two Dimensional ◽  
Cauchy Problems ◽  
Physical Point ◽  
Dimensional Lattice ◽  
The Cauchy Problem

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption. Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

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.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

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