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

We consider the Cauchy problem for a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$, with a power type, \emph{non} gauge invariant nonlinearity $\sim|u|^{p}$. We prove several ill-posedness and blowup results for both large and small $H^{s}$ data. In particular we prove that: for (essentially arbitrary) large data in $H^{\frac n2+}(\R ^n)$ the solution blows up in a finite time; for suitable large $H^{s}(\R ^n)$ data and $s< \frac{n}{2}-\frac{1}{p-1}$ no weak solution exist; when $1<p<1+\frac1n$ (or $1<p<1+\frac2n$ in $n=1,2,3$), there exist arbitrarily small initial data data for which the solution blows up in a finite time.