Blowup and ill-posedness results for a Dirac equation without gauge invariance | Zendy

Mamoru Okamoto | Zendy; Piero DʼAncona | Zendy

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Blowup and ill-posedness results for a Dirac equation without gauge invariance

Author(s) -

Mamoru Okamoto,

Piero DʼAncona

Publication year - 2016

Publication title -

evolution equations and control theory

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.665

H-Index - 19

eISSN - 2163-2480

pISSN - 2163-2472

DOI - 10.3934/eect.2016002

Subject(s) - invariant (physics) , mathematical physics , dirac (video compression format) , gauge (firearms) , nonlinear system , initial value problem , physics , cauchy distribution , cauchy problem , gauge theory , dirac equation , mathematics , mathematical analysis , quantum mechanics , archaeology , neutrino , history

We consider the Cauchy problem for\ud a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$,\ud with a power type, \emph{non} gauge invariant nonlinearity $\sim|u|^{p}$.\ud We prove several ill-posedness and blowup results for\ud both large and small $H^{s}$ data. In particular we prove that:\ud for (essentially arbitrary) large data in $H^{\frac n2+}(\R ^n)$\ud the solution blows up in a\ud finite time;\ud for suitable large $H^{s}(\R ^n)$ data and\ud $s< \frac{n}{2}-\frac{1}{p-1}$ no weak solution exist;\ud when $1

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