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DCDS

In this paper the global well-posedness in $L^2$ and $H^m$ of the Cauchy problem
is proved for nonlinear Schrödinger-type equations. This we do by establishing
regular Strichartz estimates for the corresponding linear equations and some nonlinear
a priori estimates in the framework of Besov spaces. We further establish
the regularity of the $H^m$-solution to the Cauchy problem.

keywords:
Strichartz estimates
,
Besov spaces
,
Schrödinger-type equations
,
Cauchy problem.
,
Well-posedness

DCDS

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

DCDS-S

In this paper, we consider the question of the global well-posedness
and scattering for the cubic Klein-Gordon equation $u_{t t}-\Delta
u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution
$u$ is apriorily bounded in the critical Sobolev space, that is,
$(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\mathbb{R}^d)\times H_x^{s_c-1}(\mathbb{R}^d))$
with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The
impetus to consider this problem stems from a series of recent works
for the energy-supercritical nonlinear wave equation and nonlinear
Schrödinger equation. However, the scaling invariance is broken in
the Klein-Gordon equation. We will utilize the concentration
compactness ideas to show that the proof of the global
well-posedness and scattering is reduced to disprove the existence
of the scenario: soliton-like solutions. And such solutions are
precluded by making use of the Morawetz inequality, finite speed of
propagation and concentration of potential energy.

## Year of publication

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