Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

Hidano, Kunio; Wang, Chengbo

doi:10.1007/s00029-019-0460-4

Cited by 13 publications

(8 citation statements)

References 34 publications

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“…are well-known and widely studied by scholars, see [2,3,12,14,17] and references therein. The operator…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Multiplicity and asymptotics of standing waves for the energy critical half-wave

Luo¹,

Yang²,

Xiaolong³

2021

Preprint

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In this paper, we consider the multiplicity and asymptotics of standing waves with prescribed mass R N u 2 = a 2 to the energy critical half-wavewhere N ≥ 2, a > 0, q ∈ 2, 2 + 2 N , 2 * = 2N N −1 and λ ∈ R appears as a Lagrange multiplier. We show that (0.1) admits a ground state u a and an excited state v a , which are characterised by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Several asymptotic properties of {u a }, {v a } are obtained and it is worth pointing out that we get a precise description of {u a } as a → 0 + without needing any uniqueness condition on the related limit problem. The main contribution of this paper is to extend the main results in J. Bellazzini et al. [Math. Ann. 371 (2018), 707-740] from energy subcritical to energy critical case. Furthermore, these results can be extended to the general fractional nonlinear Schrödinger equation with Sobolev critical exponent, which generalize the work of H. J. Luo-Z. T. Zhang [Calc. Var. Partial Differ. Equ. 59 (2020)] from energy subcritical to energy critical case.

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“…are well-known and widely studied by scholars, see [2,3,12,14,17] and references therein. The operator…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Equation (1.1) with power-type nonlinearities has been studied in [3]. Concerning the local/global well-posedness of solutions to (1.1), please refer to [2,14]. The dynamical properties of blow-up solutions have been investigated in e.g.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Luo¹,

Yang²,

Xiaolong³

2021

Preprint

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“…where u is a real-valued function. Subsittuting the ansatz ( 12) in (10) and taking real and imaginary parts, we have…”

Section: Note Thatmentioning

confidence: 99%

“…where R Jφ, φ dx = R |ξ||F(φ)| 2 dξ. The initial value problem of the equation (3) was studied in [1,2,10,13]. The existence of ground state and its stability was investigated in [2,4].…”

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confidence: 99%

A special form of solution to half-wave equations

Huh

2022

EECT

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“…They also play an important role in the study of well-posedness of the Cauchy problem for the nonlinear half wave equation and the nonlinear elastic wave equation with low-regularity data. See Hidano-Wang [9], Hidano-Zha [13].…”

Section: 2mentioning

confidence: 99%

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

et al. 2019

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We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data:It has been known that the problem is well-posed for s ≥ 2 and ill-posed for s < 3/2. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for s > 3/2 and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.1.1. Quadratic semi-linear wave equation . In [19], by using bilinear estimates, together with standard energy-type estimates, Klainerman and Machedon proved the local well-posedness for the problem (1.1) with a + b = 0 (thus, satisfying the null condition) in H 2 , which was later improved to H s for s > s c , see [20], [21], [22] and references therein. Moreover, even without the null condition, their bilinear

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Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

Cited by 13 publications

References 34 publications

Multiplicity and asymptotics of standing waves for the energy critical half-wave

Multiplicity and asymptotics of standing waves for the energy critical half-wave

A special form of solution to half-wave equations

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

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