Singular solutions of the<i>L</i><sup>2</sup>-supercritical biharmonic nonlinear Schrödinger equation

Baruch, Guy; Fibich, Gadi

doi:10.1088/0951-7715/24/6/009

Cited by 34 publications

(33 citation statements)

References 23 publications

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“…We thank Y. Cho for bringing us the reference [12] to our attention. Furthermore, we are grateful to G. Baruch, G. Fibich, and E. Mandelbaum for pointing out to their results obtained in [1, 3,2].…”

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confidence: 87%

Blowup for biharmonic NLS

Boulenger¹,

Lenzmann²

2017

Ann. Sci. École Norm. Sup.

114

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We consider the Cauchy problem for the biharmonic (i. e. fourth-order) NLS with focusing nonlinearity given bywhere 0 ă σ ă 8 for d ď 4 and 0 ă σ ď 4{pd´4q for d ě 5; and µ P R is some parameter to include a possible lower-order dispersion. In the mass-supercritical case σ ą 4{d, we prove a general result on finite-time blowup for radial data in H 2 pR d q in any dimension d ě 2. Moreover, we derive a universal upper bound for the blowup rate for suitable 4{d ă σ ă 4{pd´4q. In the mass-critical case σ " 4{d, we prove a general blowup result in finite or infinite time for radial data in H 2 pR d q. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems.In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.

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confidence: 87%

Blowup for biharmonic NLS

Boulenger¹,

Lenzmann²

2017

Ann. Sci. École Norm. Sup.

114

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“…Recently, the nonlinear biharmonic equation has been extensively studied. We refer readers to existing studies() for the subcritical case and other studies() for the critical case. Now let us briefly comment some known results of them.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

Gan

2018

Math Methods in App Sciences

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We consider the problem Δ 2 u = V(x)u p + in R N with u, Δu → 0 as |x| → + ∞, where p = N+4 N−4 , N ≥ 5, V is a positive continuous potential. Our aim is to construct high-energy solutions for this equation by applying the finite-dimensional reduction method and the penalization method. KEYWORDSbiharmonic equation, reduction method, supercriticalWhen > 0 is small, they established the relationship between the number of solutions and the profile of V, h, g. Also, without the restriction on , they obtained a multiplicity result.Recently, Guo, Tan, and Wang 7 concerned with the existence of least energy solutions for the following biharmonic equations:How to cite this article: Gan L. Construction of high-energy solutions for the supercritical biharmonic Schrödinger equation. Math Meth Appl Sci. 2019;42:883-891. https://doi.

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“…Let us consider the first term in this formula, which is given by (2.43). We can rewrite (2.43) as Then z 1 is rewritten as (4)…”

Section: Strichartz Estimates and Low Regularitymentioning

confidence: 99%

“…In the absence of the Laplacian, the fourth order NLS takes the form iu t + γ∆ 2 u + |u| p u = 0, x ∈ R n , t ∈ R (1.6) and it is called the biharmonic NLS. It was shown by [16] (see also [4] and the references therein) that all solutions of the biharmonic NLS are global if γ > 0. Moreover, it was found that p = 8 n is the critical exponent for singularity formation if γ < 0, and smallness in the mean-square sense is sufficient for global existence if p = 8 n .…”

Section: Introductionmentioning

confidence: 99%

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

Özsarı¹,

Yolcu²

2019

Communications on Pure &Amp; Applied Analysis

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We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.

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Singular solutions of theL²-supercritical biharmonic nonlinear Schrödinger equation

Cited by 34 publications

References 23 publications

Blowup for biharmonic NLS

Blowup for biharmonic NLS

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

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Singular solutions of theL2-supercritical biharmonic nonlinear Schrödinger equation

Cited by 34 publications

References 23 publications

Blowup for biharmonic NLS

Blowup for biharmonic NLS

Construction of high‐energy solutions for the supercritical biharmonic Schrödinger equation

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

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Singular solutions of theL²-supercritical biharmonic nonlinear Schrödinger equation