arXiv subjects

Nobuhito Miyake

Publications and source records attributed to Nobuhito Miyake.

6 recordsLinked to original sources

arXiv2026

Stability of the critical constant steady state of a Keller--Segel model

In this paper, we prove the asymptotic stability of the critical constant steady state for a simplified parabolic--elliptic Keller--Segel system in $\mathbb{R}^N$ ($N \ge 3$), which admits a one-parameter family of constant steady states. Although the stability threshold for constant steady states is known, the critical case has remained open. We also show that the convergence rate in the critical case differs from the rates obtained for previously studied subcritical constant steady states.

math.AP↗

arXiv2025

Initial traces and solvability of the fast diffusion equation with power-type nonlinearity

We investigate the qualitative behavior of the initial traces of nonnegative solutions to the fast diffusion equation with power-type nonlinearity. Necessary conditions for the existence of solutions to the corresponding Cauchy problem are identified. Moreover, sharp sufficient conditions for the existence of solutions are established by employing uniform local Morrey spaces and their generalizations.

math.AP↗

arXiv2025

Initial traces and solvability of porous medium equation with power nonlinearity

In this paper we study qualitative properties of initial traces of solutions to the porous medium equation with power nonlinearity, and obtain necessary conditions for the existence of solutions to the corresponding Cauchy problem. Furthermore, we establish sharp sufficient conditions for the existence of solutions to the Cauchy problem using uniformly local Morrey spaces and their variations, and identify the optimal singularities of the initial data for the solvability of the Cauchy problem.

math.AP↗

arXiv2024

A thresholding algorithm to Willmore-type flows via fourth order linear parabolic equation

We propose a thresholding algorithm to Willmore-type flows in $\mathbb{R}^N$. This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $Ω_0$. The main results of this paper demonstrate that the boundary $\partialΩ(t)$ of the new set $Ω(t)$, generated by our algorithm, is included in $O(t)$-neighborhood of $\partialΩ_0$ for small $t>0$ and that the normal velocity from $ \partialΩ_0 $ to $ \partialΩ(t) $ is nearly equal to the $L^2$-gradient of Willmore-type energy for small $ t>0 $. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.

math.AP↗

arXiv2020

Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

math.AP↗

arXiv2019

The homogenization method for topology optimization of structures: old and new

These are the lecture notes of a short course on the homogenization method for topology optimization of structures, given by Grégoire Allaire, during the "GSIS International Summer School 2018" at Tohoku University (Sendai, Japan). The goal of this course is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted in this course, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++.

math.AP↗