Open AccessBlowup and ill-posedness results for a Dirac equation without gauge invariance
Author(s) -
Piero D’Ancona,
Mamoru Okamoto
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