Sharp threshold of global well-posedness vs finite time blow-up for a class of inhomogeneous Choquard equations

2019 ◽  
Vol 60 (8) ◽  
pp. 081514 ◽  
Author(s):  
Majed Ghazi Alharbi ◽  
Tarek Saanouni
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Well Posedness ◽  
Sharp Threshold

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.

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.

Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases

2013 ◽  
Vol 56 (2) ◽  
pp. 378-387 ◽  
Author(s):  
Li Ma ◽  
Jing Wang
Keyword(s):  
Global Existence ◽  
Variational Problem ◽  
Invariant Manifold ◽  
Finite Time ◽  
Blow Up ◽  
Quantum Gases ◽  
Pitaevskii Equation ◽  
Sharp Threshold ◽  
Unstable Regime

AbstractIn this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

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.

scholarly journals Blow up of nonlinear multi-time fractional differential equations

2021 ◽  
Author(s):  
Weike Tang
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Time ◽  
Blow Up ◽  
Well Posedness ◽  
Burgers Equations ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we study the well-posedness of nonlinear multi-time fractional differential equations and show that the solutions of the system will blow up in finite time under certain assumptions. In particular, we apply the results to the nonlinear time fractional Burgers equations.

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.

scholarly journals Local Well-Posedness and Blow-Up for the Solutions to the Axisymmetric Inviscid Hall-MHD Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Eunji Jeong ◽  
Junha Kim ◽  
Jihoon Lee
Keyword(s):  
Classical Solution ◽  
Finite Time ◽  
Blow Up ◽  
Mhd Equations ◽  
Well Posedness ◽  
Regular Solutions ◽  
Regularity Problem ◽  
Time Existence ◽  
Hall Mhd

In this paper, we consider the regularity problem of the solutions to the axisymmetric, inviscid, and incompressible Hall-magnetohydrodynamics (Hall-MHD) equations. First, we obtain the local-in-time existence of sufficiently regular solutions to the axisymmetric inviscid Hall-MHD equations without resistivity. Second, we consider the inviscid axisymmetric Hall equations without fluids and prove that there exists a finite time blow-up of a classical solution due to the Hall term. Finally, we obtain some blow-up criteria for the axisymmetric resistive and inviscid Hall-MHD equations.

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