Stabilization in a chemotaxis model for tumor invasion
Kentarou Fujie Akio Ito Michael Winkler Tomomi Yokota
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    $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.
keywords: Chemotaxis tumor invasion. asymptotic behavior
The existence of time global solutions for tumor invasion models with constraints
Risei Kano Akio Ito
No abstract available.
keywords: quasi-variational inequality tumor invasion subdi erential
Experimental data for solid tumor cells: Proliferation curves and time-changes of heat shock proteins
Kazuhiko Yamamoto Kiyoshi Hosono Hiroko Nakayama Akio Ito Yuichi Yanagi
We consider a relation between proliferation of solid tumor cells and time-changes of the quantities of heat shock proteins in them. To do so, in the present paper we start to obtain some experimental data of the proliferation curves of solid tumor cells, actually, A549 and HepG2, as well as the time-changes of proteins, especially HSP90 and HSP72, in them. And we propose a mathematical model to re-create the experimental data of the proliferation curves and the time-changes of the quantities of heat shock proteins, which is described by ODE systems. Finally, we discuss a problem which exists between mitosis of solid tumor cells and time-changes of the quantities of heat shock proteins, from the viewpoint of biotechnology.
keywords: Time-change of HSP. Experimental data Proliferation curve
Generalized solutions of a non-isothermal phase separation model
Kota Kumazaki Akio Ito Masahiro Kubo
We study a non-isothermal phase separation model of the Penrose-Fife type. We introduce the notion of a generalized solution and prove its unique existence.
keywords: nonlinear parabolic PDE phase transitions
Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces
Akio Ito Noriaki Yamazaki Nobuyuki Kenmochi
Please refer to Full Text.
Global solvability of a model for grain boundary motion with constraint
Akio Ito Nobuyuki Kenmochi Noriaki Yamazaki
We consider a model for grain boundary motion with constraint. In composite material science it is very important to investigate the grain boundary formation and its dynamics. In this paper we study a phase-filed model of grain boundaries, which is a modified version of the one proposed by R. Kobayashi, J.A. Warren and W.C. Carter [18]. The model is described as a system of a nonlinear parabolic partial differential equation and a nonlinear parabolic variational inequality. The main objective of this paper is to show the global existence of a solution for our model, employing some subdifferential techniques in the convex analysis.
keywords: Grain boundary motion singular diffusion subdifferential.

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