Goro Akagi

A variational principle for doubly nonlinear evolution

The weighted energy-dissipation principle stands as a novel variational tool for the study of dissipative evolution and has already been applied to rate-independent systems and gradient flows. We provide here an example of its application to a specific yet critical doubly nonlinear equation featuring a super-quadratic dissipation.

Doubly Nonlinear Equations as Convex Minimization

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an

A resolution reduction method for multi-resolution terrain maps

\varepsilon

Stability of stationary solutions for semilinear heat equations with concave nonlinearity

-dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as

Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations

\varepsilon \to 0

Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

. The argument relies on the suitable dualization of the former analysis of [G. Akagi and U. Stefanelli, J. Funct. Anal., 260 (2011), pp. 2541--2578] and results in a considerable extension of the possible application range of the WED functional approach to nonlinear diffusion phenomena, including the Stefan problem and the porous media equation.

Weighted energy-dissipation functionals for doubly nonlinear evolution

Raster images such as raster terrain maps are commonly used in computer graphics. For rapid processing such as rendering and rapid feature extraction, rapid resolution reduction methods are required that keep the quality of huge images. This study deals with the resolution reduction methods.

Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate is proved by exploiting a super–subsolution method as well as the parabolic regularity theory. The base of our analysis relies on the linearization of the equation at each stationary solution and spectral analysis of the corresponding linearized operator. The main difficulties reside in the singularity of the linearized operator due to the concave nonlinearity.

Evolution inclusions governed by subdifferentials in reflexive Banach spaces

This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptotic profiles of vanishing solutions and their stability, the notions of stability and instability of group invariant profiles are introduced under a similarly invariant class of perturbations, and moreover, some stability criteria are exhibited and applied to symmetric domain (e.g., annulus) cases.

Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations

The stability of asymptotic profiles of solutions to the Cauchy–Dirichlet problem for fast diffusion equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Łojasiewicz–Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Łojasiewicz–Simon inequality in a different way.