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Sho Katayama

Publications and source records attributed to Sho Katayama.

9 recordsLinked to original sources
arXiv2026

Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition

We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial--boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.

math.AP↗
arXiv2026

Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.

math.AP↗
arXiv2025

Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $\mu>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^{\alpha}$, $\alpha>-2$, near $r=0$, $K(r)\sim r^{\beta}$, $\beta>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $\mu$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(\alpha)$ and Joseph-Lundgren exponent $p_{JL}(\alpha)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(\alpha) -2$.

math.AP↗
arXiv2024

Large time behavior of exponential surface diffusion flows on $\mathbb{R}$

We consider a surface diffusion flow of the form $V=\partial_s^2f(-\kappa)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $\kappa$ denotes the curvature and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$. We consider this equation for the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)\kappa$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $\kappa=0$.

math.AP↗
arXiv2024

Supercritical Lane-Emden equation on a cone with an inhomogeneous Dirichlet boundary condition

We consider the Lane-Emden equation with a supercritical nonlinearity with an inhomogeneous Dirichlet boundary condition on an infinite cone. Under suitable conditions for the boundary data and the exponent of nonlinearity, we give a complete classification of the existence/nonexistence of a solution with respect to the size of boundary data. Moreover, we give a result on the multiple existence of solutions via bifurcation theory. We also state results on Hardy-H\'enon equations on infinite cones as a generalization.

math.AP↗
arXiv2024

Fundamental solution to the heat equation with a dynamical boundary condition

We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition.

math.AP↗
arXiv2023

Semilinear elliptic problems on the half space with a supercritical nonlinearity

This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial nonnegative Radon measure as the boundary data. Under a suitable integrability assumption on the boundary data and the Joseph--Lundgren subcritical condition on the nonlinear term, we give a complete classification of the existence/nonexistence of a positive solution with respect to the size of the boundary data. Furthermore, we give a result on the existence of multiple positive solutions via bifurcation theory.

math.AP↗
arXiv2022

Supercritical Henon type equation with a forcing term

This paper is concerned with the structure of solutions to the elliptic problem for an Henon type equation with a forcing term. Under suitable assumptions on the forcing term, we give a complete classification of the existence/nonexistence of solutions to the problem.

math.AP↗