A class of nonlinear evolution equations governed by time-dependent operators of subdifferential type

Recently there are so many mathematical models which describe nonlinear phenomena. In some phenomena, the free energy functional is not convex. So, the existence-uniqueness question is sometimes difficult. In order to study such phenomena, let us introduce the new class of abstract nonlinear evolution equations governed by time-dependent operators of subdifferential type. In this paper we shall show the existence and uniqueness of solution to nonlinear evolution equations with time-dependent constraints in a real Hilbert space. Moreover we apply our abstract results to a parabolic variational inequality with time-dependent double obstacles constraints

Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials

We study a nonlinear evolution equation associated with time-dependent subdifferential in a separable Hilbert space. In particular, we consider an asymptotically periodic system, which means that time-dependent terms converge to time-periodic terms as time approaches infinity. Then we consider the large-time behavior of solutions without uniqueness. In such a situation the corresponding dynamical systems are multivalued. In fact, we discuss the stability of multivalued semiflows from the view-point of attractors. Namely, the main object of this paper is to construct a global attractor for asymptotically periodic multivalued dynamical systems, and to discuss the relationship to one for the limiting periodic systems

Stability for asymptotically periodic multivalued dynamical systems generated by double obstacle problems

In this paper let us consider double obstacle problems, which includes regional economic growth models. By prescribed time-dependent obstacles, our problems are nonautonomous systems and it is impossible to show the uniqueness of solutions. Therefore the associated dynamical systems are multivalued. In this paper from the viewpoint of attractors we shall consider the periodic stability for the double obstacle problem with asymptotically periodic data. Namely, assuming that time-dependent data converges to time-periodic ones as time goes to infinity, we shall construct the global attractor for the asymptotically periodic multivalued dynamical system. Moreover we shall discuss the relationship to the attractor for the limiting periodic problem

Doubly nonlinear evolution equation associated with elliptic-parabolic free boundary problems

We study an abstract doubly nonlinear evolution equations 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

Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation

We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data

Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem

Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem

Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier

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