Remarks on global solutions to the Cauchy problem for semirelativistic equations with power type nonlinearity

Fujiwara, Kazumasa; Ozawa, Tohru

doi:10.12988/ijma.2015.58211

Cited by 13 publications

(20 citation statements)

References 17 publications

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“…The author and Ozawa showed the global nonexistence in

L^{1} false(double-struckR false)

scaling critical and subcritical cases.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of the present paper is to give an alternative relatively direct proof of Propositions and . In Ikeda and Inui and Fujiwara and Ozawa, the nonexistence of weak solutions is shown by a test function method introduced by Baras‐Pierre and Zhang . However, standard test function method is not applicable to because the method relies on pointwise control of derivative of test functions.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, it is impossible to have the following pointwise estimate: There exists a positive constant C such that for any

ϕ \in C_{0}^{\infty} false(R^{n} false)

false| false({false(- normalΔ false)}^{1 false/ 2} ϕ^{ℓ} false) false(x false) false| \leq C false| ϕ^{ℓ - 1} false(x false) false({false(- normalΔ false)}^{1 false/ 2} ϕ false) false(x false) false|, \forall x \in R^{n}

with ℓ > 1. To avoid from the difficulty of fractional derivative, in Ikeda and Inui and Fujiwara and Ozawa, is transformed into

\partial_{t}^{2} v - normalΔ v = - false| λ |^{2} \partial_{t} false| u |^{p},

where

v = Im false(\overset{λ}{true} u false)

. may be obtained by applying

- Im false(\overset{λ}{true} false(i \partial t - {false(- normalΔ false)}^{1 false/ 2} false) false)

to .…”

Section: Introductionmentioning

confidence: 99%

“…with ℓ > 1. To avoid from the difficulty of fractional derivative, in Ikeda and Inui 2 and Fujiwara and Ozawa, 5 (1) is transformed into…”

Section: Introductionmentioning

confidence: 99%

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A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

Fujiwara

2018

Math Methods in App Sciences

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“…The author and Ozawa showed the global nonexistence in

L^{1} false(double-struckR false)

scaling critical and subcritical cases.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Therefore, it is impossible to have the following pointwise estimate: There exists a positive constant C such that for any

ϕ \in C_{0}^{\infty} false(R^{n} false)

false| false({false(- normalΔ false)}^{1 false/ 2} ϕ^{ℓ} false) false(x false) false| \leq C false| ϕ^{ℓ - 1} false(x false) false({false(- normalΔ false)}^{1 false/ 2} ϕ false) false(x false) false|, \forall x \in R^{n}

with ℓ > 1. To avoid from the difficulty of fractional derivative, in Ikeda and Inui and Fujiwara and Ozawa, is transformed into

\partial_{t}^{2} v - normalΔ v = - false| λ |^{2} \partial_{t} false| u |^{p},

where

v = Im false(\overset{λ}{true} u false)

. may be obtained by applying

- Im false(\overset{λ}{true} false(i \partial t - {false(- normalΔ false)}^{1 false/ 2} false) false)

to .…”

Section: Introductionmentioning

confidence: 99%

“…with ℓ > 1. To avoid from the difficulty of fractional derivative, in Ikeda and Inui 2 and Fujiwara and Ozawa, 5 (1) is transformed into…”

Section: Introductionmentioning

confidence: 99%

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A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

Fujiwara

2018

Math Methods in App Sciences

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“…Both of their proof rely on the blowup alternative and the nonexistence of global weak solutions. Fujiwara-Ozawa [11] studied the nonexistence of global weak solutions to (1) with α = 1, for p ≤ 2 in one dimension by transforming it into a wave equation…”

mentioning

confidence: 99%

Blowup results for the fractional Schrödinger equation without gauge invariance

Shi

Peng

Wang

2022

DCDS-B

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<p style='text-indent:20px;'>This paper is concerned with the nonexistence of global solutions to the fractional Schrödinger equations with order <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and nongauge power type nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ |u|^p $\end{document}</tex-math></inline-formula> for any space dimensions, where <inline-formula><tex-math id="M3">\begin{document}$ \alpha\in (0, 2] $\end{document}</tex-math></inline-formula> is assumed to be any fractional numbers. A modified test function is employed to overcome some difficulties caused by the fractional operator and to establish blowup results. Some restrictions with respect to <inline-formula><tex-math id="M4">\begin{document}$ \alpha, p $\end{document}</tex-math></inline-formula> and initial data in the previous literature are removed.</p>

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Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance

Fujiwara

Ozawa

2016

J. Evol. Equ.

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Remarks on global solutions to the Cauchy problem for semirelativistic equations with power type nonlinearity

Cited by 13 publications

References 17 publications

A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

Blowup results for the fractional Schrödinger equation without gauge invariance

Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance

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