Blow-up for the focusing energy critical nonlinear Schrödinger equation with confining harmonic potential

Xing Cheng; Yanfang Gao

Displaying similar documents to “Blow-up for the focusing energy critical nonlinear Schrödinger equation with confining harmonic potential”

Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov (2013)

Annales de l’institut Fourier

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In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^{2}$ critical...

Trudinger–Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi (2015)

Journal of the European Mathematical Society

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Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty}$ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the...

Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions

Filip Ficek (2023)

Archivum Mathematicum

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Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials. ...

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

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Nonlinear Schrödinger equations (NLS) $_{a}$ with strongly singular potential ${a | x |}^{- 2}$ on a bounded domain $Ω$ are considered. If $Ω = ℝ^{N}$ and $a > - {(N - 2)}^{2} / 4$ , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a = - {(N - 2)}^{2} / 4$ is excluded because $D (P_{a (N)}^{1 / 2})$ is not equal to $H^{1} (ℝ^{N})$ , where $P_{a (N)} : = - Δ - {(N - 2)}^{2} / {(4 | x |}^{2})$ is nonnegative and selfadjoint in $L^{2} (ℝ^{N})$ . On the other hand, if $Ω$ is a smooth and bounded domain with $0 \in Ω$ , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

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We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^{p}$ and BMO of classical ℒ-square functions.

Non-generic blow-up solutions for the critical focusing NLS in 1-D

Joachim Krieger, Wilhelm Schlag (2009)

Journal of the European Mathematical Society

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We consider the $L^{2}$ -critical focusing non-linear Schrödinger equation in $1 + 1$ -d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Jean Bourgain, Aynur Bulut (2014)

Journal of the European Mathematical Society

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We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in $ℝ^{d}$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in $ℝ^{3}$ . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

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In the half-space $ℝ^{d} \times ℝ ₊$ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$ . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

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There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^{1}$ -valued function defined on the boundary of a bounded regular domain of $R^{n}$ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within...

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V (x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $ε$ and the number of positive solutions increases to $\infty$ as $ε \to 0$ . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V (x)$ . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

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We investigate the propagation of the uniform spatial Gevrey $G^{σ}$ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

On blow-up for the Hartree equation

Jiqiang Zheng (2012)

Colloquium Mathematicae

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We study the blow-up of solutions to the focusing Hartree equation $i u_{t} + Δ u + {(| x |}^{- γ} * | u | ²) u = 0$ . We use the strategy derived from the almost finite speed of propagation ideas devised by Bourgain (1999) and virial analysis to deduce that the solution with negative energy (E(u₀) < 0) blows up in either finite or infinite time. We also show a result similar to one of Holmer and Roudenko (2010) for the Schrödinger equations using techniques from scattering theory.

Semiclassical states for weakly coupled nonlinear Schrödinger systems

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Blow-up for the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential

Jing Lu (2013)

Applicationes Mathematicae

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We give a sufficient condition under which the solutions of the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential blow up. The method is a modified variational approach, in the spirit of the work by Ibrahim et al. [Anal. PDE 4 (2011), 405-460].

Positive solutions for nonlinear Schrödinger equations with deepening potential well

Zhengping Wang, Huan-Song Zhou (2009)

Journal of the European Mathematical Society

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A critical growth rate for harmonic and subharmonic functions in the open ball in $R^{n}$

Krzysztof Samotij (1987)

Colloquium Mathematicae

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On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra, Lotfi Riahi (2004)

Annales Polonici Mathematici

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We consider the general Schrödinger operator $L = d i v (A (x) \nabla_{x}) - μ$ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function $G_{Δ}$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K ₙ^{\infty}$ considered by Zhao and Pinchover. As an application we study the cone $_{L} (ℝ ⁿ ₊)$ of all positive L-solutions continuously...

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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The main result of the present paper is : every separately-subharmonic function $u (x, y)$ , which is harmonic in $y$ , can be represented locally as a sum two functions, $u = u^{*} + U$ , where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x, y)$ outside of some nowhere dense set $S$ .

A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio, David Ruiz (2015)

Journal of the European Mathematical Society

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This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $- Δ u (x) + (ω + \frac{h^{2} (| x |)}{{| x |}^{2}} + \int_{| x |}^{+ \infty} \frac{h (s)}{s} u^{2} (s) d s) u (x) = {| u (x) |}^{p - 1} u (x)$ , where $h (r) = \frac{1}{2} \int_{0}^{r} s u^{2} (s) d s$ . This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p \in (1, 3)$ , the functional may be bounded from below or not, depending on $ω$ . Quite surprisingly, the threshold value for $ω$ is explicit....