Well-posedness of the modified dispersion-generalized Benjamin-Ono equations in modulation spaces

2021 ◽  
pp. 125933
Author(s):  
Zihua Guo ◽  
Chunyan Huang
Keyword(s):  
Modulation Spaces ◽  
Well Posedness

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 Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3145
Author(s):  
Divyang G. Bhimani ◽  
Saikatul Haque
Keyword(s):  
Initial Data ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Wiener Amalgam Spaces ◽  
General Initial Data ◽  
Amalgam Spaces ◽  
Loss Of Regularity ◽  
Bbm Equation

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

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 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.

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