arXiv subjects

Ikki Fukuda

Publications and source records attributed to Ikki Fukuda.

14 recordsLinked to original sources

arXiv2026

Asymptotic profile of solutions to the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with anisotropic dissipation in 2D

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with the dissipation term $-\nu u_{xx}$ in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropy. In this paper, we investigate the large time behavior of the solution to this problem. Especially, we show that the $L^{\infty}$-norm of the solution decays at the rate of $t^{-7/4}$ if the initial data $u_{0}(x, y)$ satisfies $(1+|x|)u_{0}\in L^{1}(\mathbb{R}^{2})$ with the zero-mass condition and some appropriate regularity assumptions. Moreover, combining techniques used for parabolic equations and the Schr\"{o}dinger equation, we also derive the detailed asymptotic profile of the solution.

math.AP↗

arXiv2025

Higher-order asymptotic profiles of solutions to the Cauchy problem for the convection-diffusion equation with variable diffusion

We consider the asymptotic behavior of solutions to the convection-diffusion equation: \[ \partial_t u - \mathrm{div}\left(a(x)\nabla u\right) = d\cdot\nabla \left(\left\lvert u\right\rvert ^{q-1}u\right),\ \ x\in\mathbb{R}^n, \ t>0 \] with an integrable initial data $u_{0}(x)$, where $n\ge1$, $q>1+\frac{1}{n}$ and $d\in \mathbb{R}^{n}$. Moreover, we take $a(x)=1+b(x)>0$, where $b(x)$ is smooth and decays fast enough at spatial infinity. It is known that the asymptotic profile of the solution to this problem can be given by the heat kernel. Moreover, some higher-order asymptotic expansions of the solution have already been studied. In particular, the structures of the second asymptotic profiles strongly depend on the nonlinear exponent $q$. More precisely, these profiles have different decay orders in each of the following three cases: $1+\frac{1}{n} 1+\frac{2}{n}$. In this paper, we focus on the critical case $q=1+\frac{2}{n}$. By analyzing the corresponding integral equation in details, we have succeeded to give the more higher-order asymptotic expansion of the solution, which generalizes the previous works.

math.AP↗

arXiv2025

Higher-order asymptotic expansions for Laplace's integral and their error estimates

We deal with the asymptotic analysis for Laplace's integral. For this problem, the so-called Laplace's method by P.S. Laplace (1812) is well-known and it has been developed in various forms over many years of studies. In this paper, we derive some formulas of the higher-order asymptotic expansions for that integral, with error estimates, which generalize previous results. Moreover, we discuss a comparison of these asymptotic formulas with approximations using numerical integral.

math.CA↗

arXiv2025

Higher-order asymptotic expansion with error estimate for the multidimensional Laplace-type integral under perturbations

We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous results including Laplace's method. The key points of the proof are a precise asymptotic analysis based on a lot of detailed Taylor expansions, and a careful consideration of the effects of the perturbations on the Hessian matrix of the phase function.

math.CA↗

arXiv2025

Mathematical and numerical analysis for some nonlinear second-order ordinary differential equations of Duffing type

In this paper, we consider the initial value problem for some nonlinear second-order ODEs of Duffing type. We study the large time behavior of the solutions to this problem, from both the perspectives of mathematical and numerical analysis. First, we derive the decay estimate of the solutions, by using the energy method. Moreover, we numerically investigate the large time behavior of the energy function related to this problem, by using a structure-preserving difference method.

math.CA↗

arXiv2024

Higher-order asymptotic profiles for solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity

We consider the asymptotic behavior of solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity. It is known that the leading term of the asymptotic profile for the solution to this problem is the Gaussian. Moreover, by analyzing the corresponding integral equation, the higher-order asymptotic expansion for the solution to the linear part and the first asymptotic profile for the Duhamel term have already been obtained. In this paper, we construct the second asymptotic profile for the Duhamel term and give the more detailed higher-order asymptotic expansion of the solutions, which generalizes the previous works. Furthermore, we emphasize that the newly obtained higher-order asymptotic profiles have a good structure in the sense of satisfying the parabolic self-similarity.

math.AP↗

arXiv2024

Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D

We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term $-μu_{xx}$. In this paper, we prove that the solution to this problem decays at the rate of $t^{-\frac{3}{4}}$ in the $L^{\infty}$-sense, provided that the initial data $u_{0}(x, y)$ satisfies $u_{0}\in L^{1}(\mathbb{R}^{2})$ and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the $L^{\infty}$-norm of the solution. As a result, we prove that the given decay rate $t^{-\frac{3}{4}}$ of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr$\ddot{\mathrm{o}}$dinger equation, we derive the explicit asymptotic profile for the solution.

math.AP↗

arXiv2023

Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev--Petviashvili--Burgers equation in 2D

We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data $u_{0}$, especially the condition $\partial_{x}^{-1}u_{0} \in L^{1}(\mathbb{R}^{2})$, it is known that the solution to this problem decays at the rate of $t^{-\frac{7}{4}}$ in the $L^{\infty}$-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at $t\to \infty$. Moreover, we obtain a lower bound of the $L^{\infty}$-norm of the solution and prove that the decay rate $t^{-\frac{7}{4}}$ of the solution given in the previous work to be optimal.

math.AP↗

arXiv2023

Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipation

We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the solutions to this problem. We prove that the solution to this problem converges to the modified heat kernel. Moreover, we construct the second term of asymptotics for the solutions depending on the degree of the nonlinearity. In view of those second asymptotic profiles, we investigate the effects of the dispersion, dissipation and nonlinear terms on the asymptotic behavior of the solutions.

math.AP↗

arXiv2023

Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

We consider the large time behavior of the solutions to the Cauchy problem for the BBM-Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.

math.AP↗

arXiv2020

Higher-order asymptotic profiles of the solutions to the viscous Fornberg-Whitham equation

We consider the initial value problem for the viscous Fornberg-Whitham equation which is one of the nonlinear and nonlocal dispersive-dissipative equations. In this paper, we establish the global existence of the solutions and study its asymptotic behavior. We show that the solution to this problem converges to the self-similar solution to the Burgers equation called the nonlinear diffusion wave, due to the dissipation effect by the viscosity term. Moreover, we analyze the optimal asymptotic rate to the nonlinear diffusion wave and the detailed structure of the solution by constructing higher-order asymptotic profiles. Also, we investigate how the nonlocal dispersion term affects the asymptotic behavior of the solutions and compare the results with the ones of the KdV-Burgers equation.

math.AP↗

arXiv2020

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profiles. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.

math.AP↗

arXiv2019

Asymptotic behavior of solutions to the generalized KdV-Burgers equation with slowly decaying data

We consider the asymptotic behavior of the global solutions to the initial value problem for the generalized KdV-Burgers equation. It is known that the solution to this problem converges to a self-similar solution to the Burgers equation called a nonlinear diffusion wave. In this paper, we derive the optimal asymptotic rate to the nonlinear diffusion wave when the initial data decays slowly at spatial infinity. In particular, we investigate that how the change of the decay rate of the initial value affects the asymptotic rate to the nonlinear diffusion wave.

math.AP↗

arXiv2019

Asymptotic stability and stability switching for a system of delay differential equations

In this paper, we consider the asymptotic stability for a system of linear delay differential equations. By analysing of the characteristic equation in detail, we have established the necessary and sufficient condition for the asymptotic stability for the zero solution of the system including the stability switching which describe the transition between stability and instability.

math.DS↗