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### Open Access Journals

DCDS

We study the global well-posedness (GWP) and small data scattering
of radial solutions of the semirelativistic Hartree type equations
with nonlocal nonlinearity $F(u) = \lambda (|\cdot|^{-\gamma}$
* $|u|^2)u$, $\lambda \in \mathbb{R}
\setminus \{0\}$, $0 < \gamma < n$, $n \ge 3$. We establish a
weighted $L^2$ Strichartz estimate applicable to non-radial
functions and some fractional integral estimates for radial
functions.

CPAA

We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.

DCDS-S

We consider the semi-relativistic Hartree type equation with
nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 <
\gamma < n, n \ge 1$. In [2, 3], the global
well-posedness (GWP) was shown for the value of $\gamma \in (0,
\frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n
\ge 3$ with small data. In this paper, we extend the previous GWP
result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with
radially symmetric large data. Solutions in a weighted Sobolev space
are also studied.

DCDS

We study the existence and scattering of global small amplitude
solutions to generalized Boussinesq (Bq) and improved modified
Boussinesq (imBq) equations with nonlinear term $f(u)$ behaving as a
power $u^p$ as $u \to 0$ in $\mathbb{R}^n, n \ge 1$.

DCDS

We study the low regularity well-posedness of the 1-dimensional
cubic nonlinear fractional Schrödinger equations with Lévy
indices $1 < \alpha < 2$. We consider both non-periodic and periodic
cases, and prove that the Cauchy problems are locally well-posed in
$H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear
estimate in Bourgain's $X^{s,b}$ space. We also show that
non-periodic equations are ill-posed in $H^s$ for $\frac {2 -
3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow
map is not locally uniformly continuous.

CPAA

In this note we study Hartree type equations with $|\nabla|^\alpha (1 < \alpha \le 2)$ and potential whose Fourier transform behaves like $|\xi|^{-(d-\gamma_1)}$ at the origin and $|\xi|^{-(d-\gamma_2)}$ at infinity. We show non-existence of scattering when $0 < \gamma_1 \le 1$ and small data scattering in $H^s$ for $s > \frac{2-\alpha}2$ when $2 < \gamma_1 \le d$ and $0 < \gamma_2 \le 2$. For this we use $U^p-V^p$ space argument and Strichartz estimates.

CPAA

In this paper we consider the initial value problem for $i\partial_t u + \omega(|\nabla|) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].

DCDS

The global well-posedness on the Cauchy problem of nonlinear
Schrödinger equations (NLS) is studied for a class of critical
nonlinearity

*below*$L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.## Year of publication

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