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Yuta Wakasugi

Publications and source records attributed to Yuta Wakasugi.

At least 19 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.

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arXiv2026

Asymptotic behavior of solutions to elliptic equations in 2D exterior domains

The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $L^{p,q}$ or the weak Lebesgue space $L^{p,\infty}$ with certain conditions on the coefficients, we give natural and an almost sharp pointwise estimate of the solution at spacial infinity. The proof is based on the argument by Korobkov--Pileckas--Russo [4], in which the decay property of the solution to the vorticity equation of the two-dimensional Navier--Stokes equations was studied.

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arXiv2025

Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity

Consider wave equations with time derivative nonlinearity and time-dependent propagation speed which are generalized versions of the wave equations in the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, the de Sitter spacetime and the anti-de Sitter space time. We show lower bounds of the lifespan of solutions as well as the global existence by providing an integrability condition on the propagation speed function, which is applicable to the nonlinear wave equation in the expanding FLRW spacetime including the de Sitter spacetime. We also prove that blow-up in a finite time occurs for the generalized form of the equation in contracting universes such as the anti-de Sitter spacetime, as well as upper bounds of the lifespan of blow-up solutions.

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arXiv2025

Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.

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arXiv2024

Appearance of Strauss-type exponent in semilinear wave equations with time-dependent speed of propagation

In this paper, blowup phenomenon for the semilinear wave equation with time-dependent speed of propagation and scattering damping is considered under the smallness of initial data. Our result contains small data blowup for sub-Strauss exponent for the simplest semilinear wave equation and also the one for semilinear generalized Tricomi equation. Key ingredient is so-called test function method (developed in Ikeda--Sobajima--Wakasa [10]) with a certain conservative quantity via a special solution with the Liouville--Green (or WKB) approximation.

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arXiv2024

Revisit on global existence of solutions for semilinear damped wave equations in $\mathbb{R}^N$ with noncompactly supported initial data

In this note, we study the Cauchy problem of the semilinear damped wave equation and our aim is the small data global existence for noncompactly supported initial data. For this problem, Ikehata and Tanizawa [5] introduced the energy method with the exponential-type weight function $e^{|x|^2/(1+t)}$, which is the so-called Ikehata--Todorova--Yordanov type weight. In this note, we suggest another weight function of the form $(1+|x|^2/(1+t))^λ$, which allows us to treat polynomially decaying initial data and give a simpler proof than the previous studies treating such initial data.

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arXiv2023

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

This paper is concerned with the semilinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates, and prove the small data global existence of solutions to the damped wave equation based on the linear estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.

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 $t \to \infty$.

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arXiv2023

Liouville-type theorems for the new Taylor--Couette flow of the stationary Navier--Stokes equations

We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders. We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily a generalized Taylor-Couette flow which is a new exact solution of the Navier--Stokes equations. If, in addition, the associated pressure is bounded or periodic in the $z$-axis, then it coincides with the well-known canonical Taylor-Couette flow. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such as the fluid is indeed, necessarily axisymmetric. It is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the new exact form of the Taylor-Couette flow.

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arXiv2023

Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain

Lifespan estimates for semilinear damped wave equations of the form $\partial_t^2u-Δu+\partial_tu=|u|^p$ in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation $\partial_tv-Δv=v^2$ with the Dirichlet boundary condition and the initial condition $v(0)=\varepsilon f$, the corresponding lifespan can be estimated from below and above by $\exp(\exp(C\varepsilon^{-1}))$ with different constants $C$. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control $L^1$-type norm and a new Gagliardo--Nirenberg type estimate with logarithmic weight are introduced.

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arXiv2022

Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data

We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the $L^2$-norm of the solution. In Appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

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arXiv2022

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that the flow is periodic in $x_3$-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

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arXiv2022

Asymptotic expansion of solutions to the wave equation with space-dependent damping

We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy method with suitable supersolutions of the corresponding parabolic problem.

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arXiv2021

Blow up of solutions of semilinear wave equations related to nonlinear waves in de Sitter spacetime

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions.The blow-up condition is the same as in the accelerated expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We also show the same results for the space derivative nonlinear term.

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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↗
arXiv2021

On Glassey's conjecture for semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime

Consider nonlinear wave equations in the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey's conjecture for the time derivative nonlinearity. We also show blow-up results for the space time derivative nonlinearity.

math.AP↗
arXiv2021

Blow up of solutions of semilinear wave equations in accelerated expanding Friedmann-Lemaître-Robertson-Walker spacetime

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann-Lemaître-Robertson-Walker spacetimes. For the case of accelerated expansion, we show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions. Comparing to the case of the Minkowski spacetime, we discuss how the scale factor affects the lifespan of blow-up solutions of the equation.

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