## Journals

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- Discrete & Continuous Dynamical Systems - A
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PROC

In this paper we consider double obstacle problems including regional
economic growth models. Unfortunately, by prescribed double obstacles, our problems lose the uniqueness of solutions. So, our problems have multiple solutions for
a given initial value. Hence, the associated dynamical systems are multivalued. In
this paper we shall consider the large-time behaviour of multiple solutions from the
viewpoint of attractors. Namely, the main object of this paper is to construct the
global attractors for non-autonomous multivalued dynamical systems associated with
double obstacle problems.

PROC

We consider periodic problems of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. In this paper we assume that the given boundary obstacles change periodically in time. Then, we prove the existence, uniqueness and asymptotic stability of a periodic solution to our problem.

PROC

We study an abstract doubly nonlinear
evolution equation associated with elliptic-parabolic free
boundary problems. In this paper we show the existence and
uniqueness of solution for the doubly nonlinear evolution
equation. Moreover we apply our abstract
results to
an elliptic-parabolic free boundary problem.

PROC

We consider a vectorial nonlinear diffusion equation with
inhomogeneous terms in one-dimensional space. In this paper we
study approximating problems of singular diffusion equations with
a piecewise constant initial data. Also we consider the
relationship between the singular diffusion problem and its
approximating ones. Moreover we give some numerical experiments
for the approximating equation with inhomogeneous terms and a
piecewise constant initial data.

DCDS

We study variational inequalities for quasilinear
elliptic-parabolic equations with time-dependent constraints.
Introducing a general condition for the time-dependence of convex
sets defining the constraints, we establish theorems concerning
existence, uniqueness as well as an order property of solutions.
Some applications of the general results are given.

PROC

We consider a phase-field model of grain boundary motion with
constraint, which is a nonlinear system of Kobayashi-Warren-Carter type: a
nonlinear parabolic partial differential equation and a nonlinear parabolic variational
inequality. Recently the existence of solutions to our system was shown
in the

*N*-dimensional case. Also the uniqueness was proved in the case when the space dimensional is one and initial data are good. In this paper we study the asymptotic stability of our model without uniqueness. In fact we shall construct global attractors for multivalued semigroups (multivalued semiflows) associated with our system in the N-dimensional case.
DCDS-S

In this paper we study an optimal control problem for a singular
diffusion equation associated with total variation energy.
The singular diffusion equation is derived as an Allen-Cahn type
equation, and then the observing optimal control problem
corresponds to a temperature control problem in the solid-liquid
phase transition.
We show the existence of an optimal control for our singular
diffusion equation by applying the abstract theory.
Next we consider our optimal control problem from the view-point of numerical analysis.
In fact we consider the approximating
problem of our equation, and we show the relationship
between the original control problem and its approximating one.
Moreover we show the necessary condition of an approximating
optimal pair, and give a numerical experiment of our approximating
control problem.

PROC

Please refer to Full Text.

keywords:

PROC

We consider the Allen--Cahn equation with a constraint. Our constraint is provided by the subdifferential
of the indicator function on a closed interval, which is the multivalued function.
In this paper we give the characterization of the Lagrange multiplier for our equation. Moreover, we consider the singular
limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem.

keywords:
double obstacle
,
singular limit
,
constraint
,
Allen-Cahn equation
,
Lagrange multiplier
,
subdierential.

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