PROC
Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint
Nobuyuki Kenmochi Noriaki Yamazaki
Conference Publications 2011, 2011(Special): 824-833 doi: 10.3934/proc.2011.2011.824
We consider a phase-fi eld model of grain boundary motion with constraint, which is a nonlinear system of Kobayashi-Warren-Carter type: a nonlinear parabolic partial diff erential 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.
keywords: singular di usion equation grain boundary motion Global attractor multivalued semigroups
PROC
Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems
Noriaki Yamazaki
Conference Publications 2003, 2003(Special): 935-944 doi: 10.3934/proc.2003.2003.935
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.
keywords: multivalued dynamical systems obstacle problems. Attractor
PROC
Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles
Masahiro Kubo Noriaki Yamazaki
Conference Publications 2007, 2007(Special): 614-623 doi: 10.3934/proc.2007.2007.614
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.
keywords: Periodic stability elliptic-parabolic variational inequality boundary constraints.
PROC
Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems
Noriaki Yamazaki
Conference Publications 2005, 2005(Special): 920-929 doi: 10.3934/proc.2005.2005.920
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.
keywords: elliptic-parabolic equation Nonlinear evolution equation free boundary problem.
PROC
Approximating problems of vectorial singular diffusion equations with inhomogeneous terms and numerical simulations
Hirotoshi Kuroda Noriaki Yamazaki
Conference Publications 2009, 2009(Special): 486-495 doi: 10.3934/proc.2009.2009.486
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.
keywords: Approximating problems numerical experiments vectorial singular diffusion
PROC
Almost periodicity of solutions to free boundary problems
Noriaki Yamazaki
Conference Publications 2001, 2001(Special): 386-397 doi: 10.3934/proc.2001.2001.386
Please refer to Full Text.
keywords:
DCDS
Elliptic-parabolic variational inequalities with time-dependent constraints
Masahiro Kubo Noriaki Yamazaki
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 335-359 doi: 10.3934/dcds.2007.19.335
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.
keywords: nonlinear evolution equations. Elliptic-parabolic variational inequalities flows in porous media
DCDS-S
Optimal control problem for Allen-Cahn type equation associated with total variation energy
Takeshi Ohtsuka Ken Shirakawa Noriaki Yamazaki
Discrete & Continuous Dynamical Systems - S 2012, 5(1): 159-181 doi: 10.3934/dcdss.2012.5.159
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.
keywords: numerical experiments. Allen-Cahn type equation Optimal control problems
PROC
Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces
Akio Ito Noriaki Yamazaki Nobuyuki Kenmochi
Conference Publications 1998, 1998(Special): 327-349 doi: 10.3934/proc.1998.1998.327
Please refer to Full Text.
keywords:
PROC
Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier
Mohammad Hassan Farshbaf-Shaker Takeshi Fukao Noriaki Yamazaki
Conference Publications 2015, 2015(special): 418-427 doi: 10.3934/proc.2015.0418
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 subdi erential.

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