Well-posedness and blow-up properties for the generalized Hartree equation

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

scholarly journals On the focusing generalized Hartree equation

2020 ◽  
Vol 9999 (9999) ◽  
pp. 1-20
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Recent Progress ◽  
Blow Up ◽  
Riesz Potential ◽  
Type Equation ◽  
Hartree Equation ◽  
Well Posedness ◽  
Nonlocal Nonlinearity ◽  
Numerical Investigations

In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.

Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations

2016 ◽  
Vol 60 (2) ◽  
pp. 481-497 ◽  
Author(s):  
Tarek Saanouni
Keyword(s):  
Heat Equation ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
High Order ◽  
Energy Space ◽  
Well Posedness ◽  
Exponential Nonlinearity

AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

The equation utt − Δu = |u|p for the critical value of p

1985 ◽  
Vol 101 (1-2) ◽  
pp. 31-44 ◽  
Author(s):  
Jack Schaeffer
Keyword(s):  
Compact Support ◽  
Finite Time ◽  
Blow Up ◽  
Critical Value ◽  
Cauchy Data ◽  
Three Dimensions ◽  
Dimensional Case ◽  
Small Data ◽  
Two Dimensional ◽  
Three Space

SynopsisThe equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.

scholarly journals Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

2014 ◽  
Vol 12 ◽  
pp. 10-20
Author(s):  
Jiang Bo Zhou ◽  
Jun De Chen ◽  
Wen Bing Zhang
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Holm Equation ◽  
Special Cases ◽  
Long Time

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

Global well-posedness and blow-up for the hartree equation

2017 ◽  
Vol 37 (4) ◽  
pp. 941-948 ◽  
Author(s):  
Lingyan YANG ◽  
Xiaoguang LI ◽  
Yonghong WU ◽  
Louis CACCETTA
Keyword(s):  
Blow Up ◽  
Hartree Equation ◽  
Well Posedness

scholarly journals Well-posedness in weighted spaces for the generalized Hartree equation with p < 2

2021 ◽  
pp. 2150074
Author(s):  
Anudeep K. Arora ◽  
Oscar Riaño ◽  
Svetlana Roudenko
Keyword(s):  
Global Existence ◽  
Local Existence ◽  
Weighted Sobolev Spaces ◽  
Riesz Transform ◽  
Hartree Equation ◽  
Positive Energy ◽  
Well Posedness ◽  
Lebesgue Spaces ◽  
Positive Time ◽  
Well Posed

We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

scholarly journals Arrest of blow up for the 3D semi-relativistic Schrödinger–Poisson system with pseudo-relativistic diffusion

2015 ◽  
Vol 27 (10) ◽  
pp. 1550023
Author(s):  
W. Abou Salem ◽  
T. Chen ◽  
V. Vougalter
Keyword(s):  
Coulomb Interaction ◽  
Finite Time ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Energy Norm ◽  
Well Posedness ◽  
System Of Equations ◽  
Initial Condition ◽  
Poisson System

We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in [Formula: see text] in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in [Formula: see text]. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.

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