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### Open Access Journals

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 $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

PROC

No abstract available.

DCDS-S

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.

PROC

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.

PROC

Please refer to Full Text.

keywords:

DCDS-S

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.

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