Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A unified method for boundedness in fully parabolic chemotaxis systems
  with signal-dependent sensitivity   This paper deals with the Keller--Segel system with signal-dependent
sensitivity \begin{equation*} u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v),
\quad v_t=\Delta v + u - v, \quad x\in\Omega,\ t>0, \end{equation*} where
$\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $\chi$ is a function
satisfying $\chi(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the
case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established
global existence of bounded solutions under the condition
$K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.;
2010; 283; 1664--1673) asserted global existence of bounded solutions for
arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried
modifying the proof; nevertheless it also has a gap. It seems to be difficult
to show global existence of bounded solutions for arbitrary $K>0$. Moreover,
the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$.
The purpose of the present paper is to obtain global existence and boundedness
under more natural and proper condition for $\chi$ and to build a mathematical
bridge between the cases $k=1$ and $k>1$.

3
The categories of this paper are: physics
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph