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Hiroshi Takeda

Publications and source records attributed to Hiroshi Takeda.

16 recordsLinked to original sources
arXiv2026

On the $L^{2}$ estimates of the diffusion waves

In this paper, we investigate the long-time behavior of the $L^2$-norm of solutions to the Cauchy problem for the strongly damped wave equation on $\mathbb{R}^n$, with particular focus on the low-dimensional cases $n=1$ and $n=2$. Although the energy is dissipative, the $L^2$-norm may grow because of low-frequency effects. We compare the diffusion-wave profile of the strongly damped equation with the corresponding free-wave evolution generated by the same initial velocity. Introducing the difference operator $D(t)$ between these two evolutions, we prove that in one dimension $D(t)$ is controlled by $Ct^{1/4}\|g\|_{L^1}$, showing that the free wave remains an effective asymptotic profile. In contrast, in two dimensions $D(t)$ has a logarithmic lower bound when the mass of the initial velocity is nonzero, implying that the wave approximation fails. Corresponding estimates for the original solution are also obtained.

math.AP↗
arXiv2025

$L^{2}$-estimates for the linear elastic waves

This paper is concerned with the large time behavior of the solution to the Cauchy problem for the elastic wave equations. In particular, optimal $L^{2}$ estimates of the elastic waves are obtained in the sense that the upper and lower bounds of the $L^{2}$ norms of each component of the solution are proved for large $t$, under the minimum assumptions necessary regarding regularity with respect to initial data. The proof is based on the approximation of the solution by a smooth auxiliary function with suitable parameters.

math.AP↗
arXiv2024

Existence and stability of time periodic solutions to nonlinear elastic wave equations with viscoelastic terms

Existence and stability of time periodic solutions for nonlinear elastic wave equations with viscoelastic terms are established. The existence of the time periodic solution is proved using the spectral decomposition of the linear principal part and the Poincaré map. On the other hand, the proof of the stability of the time-periodic solutions is generally problematic due to the slow time decay induced by the time periodic solutions. Based on the regularity estimates of the time periodic solution derived from the smoothing effect of the semigroup, sharp decay properties of the perturbation from the time periodic solution are proved, which proves the stability.

math.AP↗
arXiv2023

Large time behavior for the nonlinear dissipative Boussinesq equation

In this paper, we study the nonlinear dissipative Boussinesq equation in the whole space $\mathbb{R}^n$ with $L^1$ integrable data. As our preparations, the optimal estimates as well as the optimal leading terms for the linearized model are derived by performing the WKB analysis and the Fourier analysis. Then, under some conditions on the power $p$ of nonlinearity, we demonstrate global (in time) existence of small data Sobolev solutions with different regularities to the nonlinear model by applying some fractional order interpolations, where the optimal growth ($n=2$) and decay ($n\geqslant 3$) estimates of solutions for large time are given. Simultaneously, we get a new large time asymptotic profile of global (in time) solutions. These results imply some influence of dispersion and dissipation on qualitative properties of solution.

math.AP↗
arXiv2022

Large-time asymptotic behaviors for the classical thermoelastic system

In this paper, we study the classical thermoelastic system with Fourier's law of heat conduction in the whole space $\mathbb{R}^n$ when $n=1,2,3$, particularly, asymptotic profiles for its elastic displacement as large-time. We discover optimal growth estimates of the elastic displacement when $n=1,2$, whose growth rates coincide with those for the free wave model, whereas when $n=3$ the optimal decay rate is related to the Gaussian kernel. Furthermore, under a new condition for weighted datum, the large-time optimal leading term is firstly introduced by the combination of diffusion-waves, the heat kernel and singular components. We also illustrate a second-order profile of solution by diffusion-waves with weighted $L^1$ datum as a by-product. These results imply that wave-structure large-time behaviors hold only for the one- and two-dimensional thermoelastic systems.

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arXiv2022

Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow

In the classical theory of acoustic waves, Blackstock's model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate large-time asymptotic behaviors of the linear Cauchy problem for general Blackstock's model (that is, without Becker's assumption on monatomic perfect gases). We derive first- and second-order asymptotic profiles of solution as $t\gg1$ by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in [Chen-Ikehata-Palmieri, \emph{Indiana Univ. Math. J.} (2023)] for lower dimensional cases, but also illustrate the optimal leading term and novel second-order profiles of solution with additional weighted $L^1$ data.

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arXiv2022

Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan-Moore-Gibson-Thompson (JMGT) equation in the whole space $\mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo's law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first- and second-order asymptotic profiles of solution to the Moore-Gibson-Thompson (MGT) equation as $t\gg 1$, which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as $t\gg1$.

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arXiv2021

Smoothing effect and asymptotic behavior of solutions to nonlinear elastic wave equations with viscoelastic terms in the framework of $L^{p}$-Sobolev spaces

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12] satisfies consistency property corresponding to the additional regularity of the initial data. As a result, sharp estimates in $t$ and approximation formulas by the diffusion waves are established.

math.AP↗
arXiv2021

Global existence results for semi-linear structurally damped wave equations with nonlinear convection

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $ν(-Δ)^2 u_t$, where $ν>0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain global solutions with sharp decay properties in time. Our main purpose of this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.

math.AP↗
arXiv2017

Uniform energy decay for wave equations with unbounded damping coefficients

We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.

math.AP↗
arXiv2016

Asymptotic profiles of solutions for structural damped wave equations

In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - Δu + ν(-Δ)^σ \partial_{t} u=0$, where $ν>0$ and $0< σ\le1$. Our result is the approximation formula of the solution by a constant multiple of a special function as $t \to \infty$, which states that the asymptotic profiles of the solutions are classified into $5$ patterns depending on the values $ν$ and $σ$.

math.AP↗
arXiv2016

Large time behaivor of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. As is pointed out by [8], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Our proof is based on several estimates for the corresponding linear equations.

math.AP↗
arXiv2016

Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted $L^1$ and $L^\infty$ estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weight of the data. Our proof is based on the approximation formula of the linear solution, which is given in [36], and the nonlinear approximation theory for a nonlinear parabolic equation developed by [18].

math.AP↗
arXiv2011

Statistical Distribution of Crystallographic Groups for Inorganic Crystal Structure Database

We introduce a method that defines the species (representatives) of inorganic compounds, and studied the statistical distribution of the defined species among space groups (distribution of space groups), by using ICSD (Inorganic Crystal Structure Database). Here we show that the number of formula units in a unit cell gives a natural classification to understand the statistical distribution of crystallographic groups.

stat.AP↗