DCDS-B
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Sachiko Ishida Tomomi Yokota
Discrete & Continuous Dynamical Systems - B 2013, 18(10): 2569-2596 doi: 10.3934/dcdsb.2013.18.2569
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.
keywords: blow-up. Quasilinear degenerate Keller-Segel systems
PROC
$L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems
Sachiko Ishida
Conference Publications 2013, 2013(special): 335-344 doi: 10.3934/proc.2013.2013.335
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.
keywords: Degenerate parabolic-elliptic Keller-Segel systems $L^\infty$-decay.
PROC
An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems
Sachiko Ishida
Conference Publications 2015, 2015(special): 635-643 doi: 10.3934/proc.2015.0635
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.
keywords: Boundedness iteration quasilinear Keller-Segel system degenerate.
DCDS
Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains
Sachiko Ishida
Discrete & Continuous Dynamical Systems - A 2015, 35(8): 3463-3482 doi: 10.3934/dcds.2015.35.3463
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.
keywords: chemotaxis global existence Navier-Stokes. Degenerate diffusion
PROC
Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems
Sachiko Ishida Tomomi Yokota
Conference Publications 2013, 2013(special): 345-354 doi: 10.3934/proc.2013.2013.345
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].
keywords: global existence. Quasilinear degenerate Keller-Segel systems
DCDS-B
Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
Sachiko Ishida Yusuke Maeda Tomomi Yokota
Discrete & Continuous Dynamical Systems - B 2013, 18(10): 2537-2568 doi: 10.3934/dcdsb.2013.18.2537
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.
keywords: blow-up. Quasilinear non degenerate Keller-Segel systems

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