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### Open Access Journals

DCDS-S

The more one dives into the structural details of material or life sciences problems, the more sophisticated and specific the mathematical tools needed to address these problems become. The challenges are generally twofold: On the one hand one wishes to find accurate descriptions of the microscale, while on the other hand, having in view certain microscale dynamics (close to micro phase transitions), one wishes to capture a basic understanding over much larger scales. In both cases one aims at well-posed PDE models in the sense of Hadamard, which are not only computable numerically but also verifiable against experiments.

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keywords:

DCDS

This paper is concerned with the subdifferential operator approach
to nonlinear (possibly degenerate and singular) parabolic PDE's of the form $u_t-\Delta \beta(u) \ni f$ formulated in the dual space of $H^1(\Omega)$, where $\beta$ is a maximal monotone
graph in $\mathbf R\times \mathbf R$. In the set-up considered so far [8], some coerciveness condition has
been required for $\beta$, corresponding at least to the fact that it is onto $\mathbf R$. In the
present paper, we show that the subdifferential operator approach is possible for any
maximal monotone graph $\beta$ without any growth condition.

DCDS

This paper is concerned with a heat convection problem. We
discuss it in the framework of parabolic variational inequalities.
The problem is a system of a heat equation with convection and a Navier-Stokes variational inequality with
temperature-dependent velocity constraint. Our problem is a sort of parabolic
quasi-variational inequalities in the sense that the constraint set for the velocity
field depends on the unknown temperature. We shall give an existence result of the heat convection
problem in a weak sense, and show that under some additional constraint for temperature there exists
a strong solution of the problem.

DCDS-S

In this paper,
the Navier-Stokes variational inequality with the temperature
dependent constraint is considered in 3-dimensional space.
This problem is motivated by an initial-boundary
value problem for a thermohydraulics model in which
the absolute value of the velocity field
is constrained,
depending on the unknown temperature.
The abstract theory of nonlinear evolution equations
governed by subdifferentials
of time-dependent convex functionals is useful in showing the
existence of a solution.
In the mathematical treatment,
the point of emphasis is to specify
a class of time-dependence of convex constraints.

PROC

In this paper, the existence and uniqueness questions of
abstract parabolic variational inequalities are considered in
connection with Lagrange multipliers.
The focus of authors' attention is the characterization of parabolic variational
inequalities by
means of Lagrange multipliers.
It is well-known that various kinds of parabolic differential equations under convex
constraints are represented by variational inequalities with time-dependent
constraints, and
the usage of Lagrange multipliers associated with constraints makes it possible to
reformulate the variational inequalities as equations.
In this paper, as a typical case,
a parabolic problem with nonlocal time-dependent obstacle is treated
in the framework of abstract evolution equations governed by time-dependent
subdifferentials.

DCDS-S

In this paper we consider some mechanical phenomena whose dynamics is
described by a class of quasi-variational inequalities of parabolic type.
Our system consists of a second-order parabolic variational inequality with gradient
constraint depending on the temperature and the heat equation. Since the
temperature is unknown in our problem,
the constraint function is unknown as well. In this sense, our problem includes the
quasi-variational structure, and in the mathamtical analysis one of main difficulties comes from it.
Our approach to
the problem is based on the abstract theory of quasi-variational inequalities with non-local
constraint which has been developed in [6]. However the abstract theory is not
directly used in the existence proof of a solution, since the mathematical situation of
the problem is much nicer than that in the abstract theory [6]. In this paper we prove the existence
of a weak solution of our system.

CPAA

This paper is concerned with phase-field systems of
Penrose-Fife type which model the dynamics of a phase transition
with non-conserved vectorial order parameter. The main novelty of
the model is that the evolution of the order parameter vector is
governed by a system consisting of one partial differential equation
and one partial differential inclusion, which in the simplest case
may be viewed as a diffusive approximation of the so-called
multi-dimensional stop operator, which is one of the fundamental
hysteresis operators. Results concerning existence, uniqueness and
continuous dependence on data are presented which can be viewed as
generalizations of recent results by the authors to cases where a
diffusive hysteresis occurs.

keywords:
Parabolic systems
,
phase transitions.
,
hysteresis
,
phase-ﬁeld models
,
a priori estimates
,
existence
,
uniqueness

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.
PROC

Please refer to Full Text.

keywords:

PROC

It is the main objective of this paper to discuss applications of the abstract results recently evolved in [16,17] to the solvability of variational inequalities with constraints depending on the unknown functions, which are called quasi-variational inequalities.

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