Threshold For Global Existence Research Articles

Threshold of global existence for a Gross–Pitaevskii system associated with a dipolar Bose–Einstein condensate in 2D

Threshold of global existence for a Gross–Pitaevskii system associated with a dipolar Bose–Einstein condensate in 2D

Well-posedness and blow-up of Virial type for some fractional inhomogeneous Choquard equations

ABSTRACT In the subcritical energy case, local well-posedness is established in the radial energy space for a class of fractional inhomogeneous Choquard equations. The best constant of a Gagliardo–Nirenberg type inequality is obtained. Moreover, a sharp threshold of global existence versus blow-up dichotomy is obtained for mass super-critical and energy subcritical solutions.

Mass concentration for nonlinear Schrödinger equation with partial confinement

This paper studies the dynamical properties of blow-up solutions for nonlinear Schrödinger equation with partial confinement, which may model the Bose-Einstein condensate under a partial trap potential. By using the variational characteristic of the classic nonlinear scalar field equation and the Hamilton conservations, we first get the threshold for global existence and blow-up of the Cauchy problem on mass in two-dimensional space. Then, in terms of the refined compactness Lemma and the variational characteristic of the ground state of nonlinear scalar field equation, we get mass concentration properties of the blow-up solutions as well as limiting profile of the blow-up solutions with small super-critical mass.

Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrödinger System in ℝ3

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schrodinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.

Sharp criteria for the nonlinear Schrödinger equation with combined nonlinearities of power-type and Hartree-type

This paper deals with the Cauchy problem of the nonlinear Schrödinger equation with Combined Nonlinearities. By using the generalized Gagliardo–Nirenberg inequalities, when and for all , we obtain a sharp threshold for global existence and blow-up of solutions.

The Focusing Cubic NLS on Exterior Domains in Three Dimensions

We consider the focusing cubic nonlinear Schrödinger equation (NLS) in the exterior |$\Omega $| of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if |$E(u_0)M(u_0)<E(Q)M(Q)$| and |$\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$|⁠, the corresponding solution to the initial value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, |$Q(x)$| denotes the ground state for the focusing cubic NLS in |$ \mathbb {R}^3$|⁠.

Sharp threshold of global existence for the generalized Zakharov system with three-dimensional magnetic field in the subsonic limit

The paper is concerned with the Zakharov system with three dimensional magnetic field in the subsonic limit, which describes the spontaneous generation of a magnetic field in cold plasma. In fact, we will investigate the effect of the magnetic field term on the equations. By using variational calculus, we obtain the threshold of global existence and blowup of the solution to the equations, which will be more complicated and requires more skill.

Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schr\"odinger-Poisson-Slater equations in $\R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schr\"odinger-Poisson type equation \label{evolution1} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 % \text{in} \R^{3}, when $p \in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| \triangledown u|_{L^{2}(\mathbb{R}^3)}^2+1/4\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-\frac{1}{p}\int_{\mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= \{u \in H^1(\mathbb{R}^3) :|u|_{L^2(\R^3)}^2=c, c>0}.$$ For the values $p \in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0\in H^1(\R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $\gamma(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schr\"odinger-Poisson-Slater equation and the classical nonlinear Schr\"odinger equation.

Sharp threshold of global existence for non-linear Gross–Pitaevskii equation in ℝN

This paper is concerned with the non-linear Gross-Pitaevskii equation which describes the attractive Bose-Einstein condensate under a magnetic trap. By an intricate variational argument we derive out a sharp threshold of blowing up and global existence by applying the potential well argument and the concavity method. Furthermore, we answer the question: How small are the initial data, the global solutions of the Cauchy problem of the equation exist for p > 1 + 4 N?