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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-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

This paper deals with
quasilinear degenerate Keller-Segel systems
of parabolic-elliptic type.
In this type,
Sugiyama-Kunii [10] established
the $L^r$-decay ($1\leq r<\infty$) of solutions
with small initial data
when $q\geq m+\frac{2}{N}$
($m$ denotes the intensity of diffusion and
$q$ denotes the nonlinearity).
However, the $L^\infty$-decay property was not obtained yet.
This paper gives the $L^\infty$-decay property
in the super-critical case with small initial data.

PROC

This paper mainly considers the uniform bound on
solutions of non-degenerate Keller-Segel systems on the whole space.
In the case that the domain is bounded,
Tao-Winkler (2012) proved existence of globally bounded solutions of non-degenerate systems.
More precisely, they gave the result
on boundedness in quasilinear parabolic equations
by using the $L^p$-bounds on the solution for some large $p>1$.
In Ishida-Yokota (2012), they dealt with the same system as this paper
on the whole space,
however,
their $L^\infty$-estimate possibly grows up even if
the solutions have the uniform bounds in
$L^p(\mathbb{R}^N)$ for all $p\in[1,\infty)$.
The present work asserts the uniform in time $L^\infty$-bound on solutions.
Moreover, this paper covers the degenerate Keller-Segel systems
and constructs the uniformly bounded weak solutions.

DCDS

This paper is concerned with degenerate chemotaxis-Navier-Stokes systems
with position-dependent sensitivity
on a two dimensional bounded domain.
It is known that in the case without a position-dependent sensitivity function,
Tao-Winkler (2012) constructed a globally bounded weak solution of
a chemotaxis-Stokes system with any porous medium diffusion,
and
Winkler (2012, 2014) succeeded
in proving global existence
and stabilization of classical solutions
to a chemotaxis-Navier-Stokes system with linear diffusion.
The present work shows global existence and boundedness of weak solutions
to a chemotaxis-Navier-Stokes system
with position-dependent sensitivity for any porous medium diffusion.

DCDS-S

This paper deals with the quasilinear Keller-Segel system

$ \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &x \in \Omega, \ t>0, \\ \ v_t = \Delta v - v +u, &x \in \Omega, \ t>0 \end{cases} \end{align*} $ |

in

or in a smoothly bounded domain

, with nonnegative initial data

, and

; in the case that

is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity

and the sensitivity

are assumed to fulfill

and

, respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when

. In the case

the result says boundedness which was not attained in a previous paper (

$ \Omega = \mathbb{R}^N $ |

$ \Omega\subset \mathbb{R}^N $ |

$ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $ |

$ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $ |

$ \Omega $ |

$ D(u) $ |

$ S(u) $ |

$ D(u)\ge u^{m-1}\ (m\geq1) $ |

$ S(u)\leq u^{q-1}\ (q\geq 2) $ |

$ q<m+\frac{2}{N} $ |

$ \Omega = \mathbb{R}^N $ |

*J. Differential Equations*2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (*J. Differential Equations*2012; 252:692-715) in the case of bounded domains.
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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