Ken Shirakawa

WELL-POSEDNESS AND PERIODIC STABILITY FOR QUASILINEAR PARABOLIC VARIATIONAL INEQUALITIES WITH TIME-DEPENDENT CONSTRAINTS

We study variational inequalities with time-dependent constraints for quasilinear parabolic PDE of divergence from. Introducing a general condition on the constraints, we prove existence. uniqueness and order property of solution. Some applications are given.

Energy dissipative solutions to the Kobayashi–Warren–Carter system

In this paper we study a variational system of two parabolic PDEs, called the Kobayashi–Warren–Carter system, which models the grain boundary motion in a polycrystal. The focus of the study is on the existence of solutions to this system which dissipate the associated energy functional. We obtain the existence of this type of solution via a suitable approximation of the energy functional with Laplacians and an extra regularization of the weighted total variation term of the energy. As a byproduct of this result, we also prove some -convergence results concerning weighted total variations and the corresponding time-dependent cases. Finally, the regularity obtained for the solutions together with the energy dissipation property, permits us to completely characterize the ω-limit set of the solutions.

Weak Formulation for Singular Diffusion Equation with Dynamic Boundary Condition

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.

Solvability for a PDE Model of Regional Economic Trend

The aim of this work is to develop a simulation method focused on regional economic trend. In this light, an original model, formulated by partial differential equations, will be proposed. Consequently, the existence of time-local solutions of our mathematical model will be concluded, as a transitional report in the research.