arXiv subjects

Yuki Tsukamoto

Publications and source records attributed to Yuki Tsukamoto.

8 recordsLinked to original sources

arXiv2026

Delayed diffusion with measure-valued kernels in nonlinear parabolic equations

We study nonlinear parabolic equations with delayed diffusion terms governed by finite signed measure kernels. The atom of the kernel at the origin is absorbed into the present-time operator, while the remaining part is treated as a residual delay kernel. Under structural assumptions on the effective present-time operators and a pathwise coercivity condition for the total memory operator, we prove the existence and uniqueness of weak solutions and their stability under weak-star convergence of the kernels. The stability result covers collapsing delayed atoms, whose mass is transferred to the present-time diffusion coefficient in the limit. We verify the assumptions for p-Laplacian type examples, including separated kernels and a regularized class of kernels reaching the origin.

math.AP↗

arXiv2026

Vanishing interfaces in an asymmetric fast reaction limit

We study the fast reaction limit for a two-component reaction-diffusion system with asymmetric reaction terms, where only one component diffuses. For nonnegative and mutually segregated initial data, we prove that the initial interface vanishes instantaneously. More precisely, the diffusive component converges uniformly to the solution of the heat equation, while the non-diffusive component vanishes away from the initial time. The proof is based on explicit barriers and a comparison argument, and applies under both Dirichlet and Neumann boundary conditions.

math.AP↗

arXiv2025

Formation of stationary interfaces in the fast reaction limit

We study a two-component reaction-diffusion system in which one of the reaction terms becomes singularly large. Assuming that the initial data are nonnegative and mutually segregated, we prove that the solution converges to that of the heat equation in the nonreactive region, while it remains zero elsewhere. The analysis is based on explicitly constructed barrier functions and a comparison argument adapted to systems with asymmetric reaction terms, which together provide control near the interface. As a result, the limiting behavior exhibits a stationary phase interface determined by the initial support of the reactive component.

math.AP↗

arXiv2023

Convergence of the Allen-Cahn equation with transport term in a bounded domain

We study the Allen-Cahn equation with respect to a transport term in a bounded domain. We prove that the limit interface is the mean curvature flow with the transport term, given the condition that the energy is uniformly bounded with respect to time. Using this result, we show the existence of the mean curvature flow with a gradient vector field as the transport term.

math.AP↗

arXiv2023

Interface disappearance in fast reaction limit

We study the singular limit problem referred to as the fast reaction limit. This problem has been extensively studied when the same reaction term is used in a two-component system. However, the behavior of the solution under different reaction terms remains not yet well understood. In this paper, we will consider the problem where the reaction term is represented by a power term. We prove that the initial interface disappears immediately, and the function converges to a solution that satisfies the heat equation.

math.AP↗

arXiv2020

Prescribed mean curvature equation on torus

Prescribed mean curvature problems on the torus has been considered in one dimension. In this paper, we prove the existence of a graph on the $n$-dimensional torus $\mathbb {T}^n$, the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

math.AP↗

arXiv2020

A diffused interface with the advection term in a Sobolev space

We study the asymptotic limit of diffused surface energy in the van der Waals--Cahn--Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min-max method.

math.AP↗

arXiv2019

The Dirichlet problem for a prescribed mean curvature equation

We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.

math.AP↗