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### Open Access Journals

PROC

The global existence of weak solutions
to quasilinear ``degenerate'' Keller-Segel systems
is shown in the recent papers [3], [4].
This paper gives some improvements and supplements
of these.
More precisely, the differentiability and the smallness
of initial data are weakened
when the spatial dimension $N$ satisfies $N\geq2$.
Moreover, the global existence is established
in the case $N=1$
which is unsolved in [4].

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-B

This paper gives a blow-up result for
the quasilinear degenerate Keller-Segel systems
of parabolic-parabolic type.
It is known that
the system has a global solvability in the case where $q < m + \frac{2}{N}$
($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity)
without any restriction on the size of initial data,
and where $q \geq m + \frac{2}{N}$ and the initial data are ``small''.
However, there is no result when $q \geq m + \frac{2}{N}$
and the initial data are ``large''.
This paper discusses such case and
shows that
there exist blow-up energy solutions
from initial data having large negative energy.

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

This paper deals with the chemotaxis system
\[
\begin{cases}
u_t=\Delta u - \nabla \cdot (u\nabla v),
\qquad x\in \Omega, \ t>0, \\
v_t=\Delta v + wz,
\qquad x\in \Omega, \ t>0, \\
w_t=-wz,
\qquad x\in \Omega, \ t>0, \\
z_t=\Delta z - z + u,
\qquad x\in \Omega, \ t>0,
\end{cases}
\]
in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$,
that has recently been proposed as a model for tumor invasion
in which the role of an active extracellular matrix is accounted for.

It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$,

$u(x,t)\to \overline{u_0}$ , $v(x,t) \to \overline{v_0} + \overline{w_0}$, $w(x,t) \to 0$ and $z(x,t) \to \overline{u_0}$, uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ and $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$,

$u(x,t)\to \overline{u_0}$ , $v(x,t) \to \overline{v_0} + \overline{w_0}$, $w(x,t) \to 0$ and $z(x,t) \to \overline{u_0}$, uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ and $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

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].

PROC

This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic
Keller-Segel system
$u_t
= \nabla \cdot(D(u)\nabla u)
-\nabla \cdot (v^{-1}S(u)\nabla v)$,
$v_t= \Delta v-v+u$.
In [7,4], global existence and boundedness were established in the system without $v^{-1}$.
In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law.
A

*uniform-in-time*estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.
DCDS-B

Gradient estimate for solutions
to quasilinear non-degenerate Keller-Segel systems
on $\mathbb{R}^N$

This paper gives the gradient estimate
for solutions to the quasilinear
non-degenerate parabolic-parabolic
Keller-Segel system (KS)
on the whole space $\mathbb{R}^N$.
The gradient estimate for (KS) on bounded domains
is known as an application of Amann's existence theory
in [1].
However, in the whole space case
it seems necessary to derive the gradient estimate directly.
The key to the proof is a modified Bernstein's method.
The result is useful to obtain the whole space version of
the global existence result by Tao-Winkler [13] except for the boundedness.

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