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### Open Access Journals

DCDS

The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on
$\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and
$b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$,
while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$.
Over twenty years ago late Professor Kato proved that the minimal
realization $T_{min}$ is essentially quasi-$m$-accretive in
$L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in
L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is
shown that under some additional conditions the same conclusion
remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.

DCDS-S

Existence of unique strong solutions is established for
Schrödinger type evolution equations with
monotone nonlinearity.
The proof is based on a perturbation theorem for
$m$-accretive operators in a complex Hilbert space.

PROC

Please refer to Full Text.

DCDS

Two theorems concerning strong wellposedness are established
for the complex Ginzburg-Landau equation.
One of them is concerned with strong $L^{2}$-wellposedness,
that is,
strong wellposedness for $L^{2}$-initial data.
The other deals with
$H_{0}^{1}$-initial data as a partial extension.
By a technical innovation it becomes possible to prove the
convergence of approximate solutions without compactness.
This type of convergence is known with accretivity methods
when the argument of the complex coefficient
is small.
The new device yields the generation of a class
of non-contraction semigroups even when the argument is
large.
The results are both obtained as application of abstract theory
of semilinear evolution equations with subdifferential operators.

DCDS-S

Let $X$ be a complex Banach space and
$A:\,D(A) \to X$ a quasi-$m$-sectorial operator
in $X$. This paper is concerned with the
identification of diffusion coefficients
$\nu > 0$ in the initial-value problem:
\[
(d/dt)u(t) + {\nu}Au(t) = 0,
\quad t \in (0,T), \quad u(0) = x \in X,
\]
with additional condition $\|u(T)\| = \rho$,
where $\rho >0$ is known. Except for
the additional condition, the solution to the
initial-value problem is given by
$u(t) := e^{-t\,{\nu}A} x
\in C([0,T];X) \cap C^{1}((0,T];X)$.
Therefore, the identification of $\nu$ is reduced
to solving the equation
$\|e^{-{\nu}TA}x\| = \rho$.
It will be shown that the unique root
$\nu = \nu(x,\rho)$
depends on $(x,\rho)$ locally Lipschitz
continuously if the datum $(x,\rho)$ fulfills
the restriction $\|x\|> \rho$. This extends
those results in
Mola [6](2011).

DCDS-S

A new existence and uniqueness theorem is established for
linear evolution equations of hyperbolic type with
strongly measurable coefficients in a separable Hilbert space.
The result is applied to
the Dirac equation with time-dependent potential.

EECT

So far there seems to be no abstract formulations
for nonlinear Schrödinger equations (NLS).
In some sense
Cazenave[2, Chapter 3] has given a guiding principle to replace the
free Schrödinger group with the approximate identity of resolvents.
In fact, he succeeded in separating the existence theory
from the Strichartz estimates.
This paper is a proposal
to extend his guiding principle by using the square root of the resolvent.
More precisely,
the abstract theory here unifies the
local existence of weak solutions to (NLS) with
not only typical nonlinearities
but also some critical cases.
Moreover, the theory yields the improvement
of [21].

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