scholarly journals Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory

2009 ◽  
Author(s):  
T. Oh
Keyword(s):  
Type Systems ◽  
Local Theory ◽  
Well Posedness

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 multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Numerical Approximation ◽  
Method Of Lines ◽  
Type Systems ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Model Surface ◽  
Hamiltonian Structures ◽  
Surface Wave Propagation ◽  
The Method Of Lines

© 2019 Elsevier B.V. In this paper we consider the numerical approximation of systems of BOUSSINESQ-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and HAMILTONIAN formulations, well-posedness and existence of solitary-wave solutions)were previously analysed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and HAMILTONIAN structures, of different strategies for the spatial and time discretizations are discussed and illustrated.

scholarly journals On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Numerical Approximation ◽  
Method Of Lines ◽  
Type Systems ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Model Surface ◽  
Hamiltonian Structures ◽  
Surface Wave Propagation ◽  
The Method Of Lines

© 2019 Elsevier B.V. In this paper we consider the numerical approximation of systems of BOUSSINESQ-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and HAMILTONIAN formulations, well-posedness and existence of solitary-wave solutions)were previously analysed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and HAMILTONIAN structures, of different strategies for the spatial and time discretizations are discussed and illustrated.

scholarly journals Global well-posedness of logarithmic Keller-Segel type systems

2021 ◽  
Vol 287 ◽  
pp. 185-211
Author(s):  
Jaewook Ahn ◽  
Kyungkeun Kang ◽  
Jihoon Lee
Keyword(s):  
Type Systems ◽  
Well Posedness

scholarly journals Well-posedness and stability for Bresse-Timoshenko type systems with thermodiffusion effects and nonlinear damping

2021 ◽  
Vol 6 (3) ◽  
pp. 2704-2721
Author(s):  
Khaled zennir ◽  
◽  
Djamel Ouchenane ◽  
Abdelbaki Choucha ◽  
Mohamad Biomy ◽  
...  
Keyword(s):  
Type Systems ◽  
Nonlinear Damping ◽  
Well Posedness ◽  
Timoshenko Type

A local theory for a fractional reaction-diffusion equation

2019 ◽  
Vol 21 (06) ◽  
pp. 1850033
Author(s):  
Arlúcio Viana
Keyword(s):  
Diffusion Equation ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Local Theory ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Fractional Reaction

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

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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