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### Open Access Journals

EECT

We consider the Cauchy problem for
a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$,
with a power type,

*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 $H^{\frac n2+}(\mathbb{R} ^n)$ the solution blows up in a finite time; for suitable large $H^{s}(\mathbb{R} ^n)$ 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.
keywords:
blow up
,
Dirac equation
,
non gauge invariance
,
$H^s$-solution.
,
nonexistence of solution

DCDS

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\mathbb{R} ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$.
We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.

CPAA

In the present paper, we consider the Cauchy problem of fourth order nonlinear
Schrödinger type equations with derivative nonlinearity.
In one dimensional case, the small data global well-posedness and scattering for the fourth order nonlinear Schrödinger equation with the nonlinear term $\partial _x (\overline{u}^4)$ are shown in the
scaling invariant space $\dot{H}^{-1/2}$.
Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$, where $d$ denotes the space dimension.

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