arXiv subjects

Shuji Yoshikawa

Publications and source records attributed to Shuji Yoshikawa.

5 recordsLinked to original sources

arXiv2026

Asymptotic profiles for the Cauchy problem of semilinear beam equation with two variable coefficients in the subcritical case

In this article, we investigate the asymptotic profile of solutions to the Cauchy problem for a nonlinear beam equation with two variable coefficients in the subcritical nonlinear case. In contrast to our previous result [6], in which the asymptotic profile is governed by the linear heat kernel and the nonlinear effect is asymptotically negligible, the asymptotic profile in the present setting is described by a self-similar solution to the associated nonlinear parabolic equation (constructed in Brezis-Peletier-Terman [1]). The proof relies on delicate energy estimates in weighted spaces formulated in parabolic self-similar variables.

math.AP↗

arXiv2023

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right). \] In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.

math.AP↗

arXiv2021

Structure-preserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions

In this article we discuss the numerical analysis for the finite difference scheme of the one-dimensional nonlinear wave equations with dynamic boundary conditions. From the viewpoint of the discrete variational derivative method we propose the derivation of the structure-preserving finite difference schemes of the problem which covers a variety of equations as widely as possible. Next, we focus our attention on the semilinear wave equation, and show the existence and uniqueness of solution for the scheme and error estimates with the help of the inherited energy structure.

math.NA↗

arXiv2020

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada (Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for the proposed scheme. Computation examples demonstrate the effectiveness of the proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.

math.NA↗

arXiv2016

Pathwise existence of solutions to the Implicit Euler method for the stochastic Cahn-Hilliard Equation

We consider the implicit Euler approximation of the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$. We show pathwise existence and uniqueness of solutions for the method under a restriction on the step size that is independent of the size of the initial value and of the increments of the Wiener process. This result also relaxes the imposed assumption on the time step for the deterministic Cahn-Hilliard equation assumed in earlier existence proofs.

math.NA↗