Searcharxiv⌕ Search

arXiv subjects

Kyouhei Wakasa

Publications and source records attributed to Kyouhei Wakasa.

At least 19 recordsLinked to original sources
arXiv2026

A revisit via slicing method on a quadratic semilinear wave equation in two space dimensions

In this paper, we are focusing on the proof of the blow-up result for a quadratic semilinear wave equation in two space dimensions. There is a logarithmic loss in estimating the lifespan of classical solutions if the 0th moment of the initial speed does not vanish. This result is already known with almost sharp constants. But in order to have a direct application to numerical analysis, we show a simple proof by iteration argument of point-wise estimate of the solution with the slicing technique.

math.AP↗
arXiv2025

Blow-up of solutions for discrete semilinear wave equation with the scale-invariant damping

We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents depending on the space dimensions. Our purpose is to obtain results for the discretized equations that correspond to those shown for the continuous one. The proof is based on Matsuya [6], who showed the blow-up problem for discrete semilinear wave equations without dissipative terms, and we found that the result is sharp in the case of one and two space dimensions compared to the continuous equations.

math.AP↗
arXiv2021

Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simple connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.

math.AP↗
arXiv2021

Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass

In the present paper, we study small data blow-up of the semi-linear wave equation with a scattering dissipation term and a time-dependent mass term from the aspect of wave-like behavior. The Strauss type critical exponent is determined and blow-up results are obtained to both sub-critical and critical cases with corresponding upper bound lifespan estimates. For the sub-critical case, our argument does not rely on the sign condition of dissipation and mass, which gives the extension of the result in \cite{Lai-Sch-Taka18}. Moreover, we show the blow-up result for the critical case which is a new result.

math.AP↗
arXiv2020

On the critical decay for the wave equation with a cubic convolution in 3D

We consider the wave equation with a cubic convolution $\partial_t^2 u-Δu=(|x|^{-γ}*u^2)u$ in three space dimensions. Here, $0<γ<3$ and $*$ stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then $γ\ge2$ assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when $2\le γ<3$. In this paper, we consider the Cauchy problem for $2\le γ<3$ in the space-time weighted $L^\infty$ space in which functions have critical decay rate. When $γ=2$, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in Kubo(2004)). When $2<γ<3$, we also prove unique global existence of solutions for small data.

math.AP↗
arXiv2020

The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed. The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between "wave-like" domain and "heat-like" domain. As a result, we obtain a new type of estimate especially for the critical exponent.

math.AP↗
arXiv2020

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in \left(-\frac{1}{2},3\right)$ in three spatial dimension for initial data $\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\times C^1(\mathbb{R}^3)$ with a compact support, where $v=v(x,t)$ is an unknown function to the problem on $\mathbb{R}^3\times[0,T)$. Here $T$ denotes a maximal existence time of $v$. The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case $γ\in (0,3)$, and show a lower estimate of the lifespan in the critical or subcritical case $γ\in \left(-\frac{1}{2},0\right]$. The essential part for their proofs is to derive a weaker estimate under the weaker condition than the case without damping and to recover the weakness by the effect of the dissipative term. The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case $γ\in \left(-\frac{1}{2},0\right)$. The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in \cite{H20} to obtain the upper estimate of the lifespan. Our two results determine that a critical exponent $γ_c$ which divides global existence and blow-up for small solutions is $0$, namely $γ_c=0$. As the result, we can see that the critical exponent shift from $2$ to $0$ due to the effect of the scale invariant damping term.

math.AP↗
arXiv2020

Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in (0,n)$, where $v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data $(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+ν)})$ as $|x|\rightarrow\infty$. Here $ν$ belongs to the scaling supercritical case $ν<\frac{n-γ}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($n\ge 4$).

math.AP↗
arXiv2019

The lifespan of solutions of semilinear wave equations with the scale-invariant damping in one space dimension

The critical constant of time-decaying damping in the scale-invariant case is recently conjectured. It also has been expected that the lifespan estimate is the same as for the associated semilinear heat equations if the constant is in the \heat-like" domain. In this paper, we point out that this is not true if the total integral of the sum of initial position and speed vanishes. In such a case, we have a new type of the lifespan estimates which is closely related to the non-damped case in shifted space dimensions.

math.AP↗
arXiv2018

Finite time blowup of solutions to semilinear wave equation in an exterior domain

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in exterior domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in $\mathbf{R}^N$. At least in the case $N=4$, their estimates are sharp in view of the work by Zha--Zhou (2015). The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa--Yordanov.

math.AP↗
arXiv2018

Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.

math.AP↗
arXiv2018

On the blow-up for critical semilinear wave equations with damping in the scattering case

We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\mathbf{R}^n$, where $n\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p<p_0(n)$. In this paper, we extend their results to the critical case $p=p_0(n)$. The proof is based on Wakasa and Yordanov (2018), which concerns the blow-up and upper bound of lifespan for critical semilinear wave equations with variable coefficients.

math.AP↗
arXiv2018

Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou (1992). However, our framework is much simpler than that. As a consequence, we found new $(p,q)$-curve for the system $\partial_t^2u-Δu=|v|^q$, $\partial_t^2v-Δv=|\partial_tu|^p$ and lifespan estimate for small solutions for new region.

math.AP↗
arXiv2018

Blow-up of solutions to critical semilinear wave equations with variable coefficients

We verify the critical case $p=p_0(n)$ of Strauss' conjecture (1981) concerning the blow-up of solutions to semilinear wave equations with variable coefficients in $\mathbf{R}^n$, where $n\geq 2$. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou(2007) and Zhou and Han (2014): exponential "eigenfunctions" of the Laplacian are used to construct the test function $ϕ_q$ for linear wave equation with variable coefficients and John's method of iterations (1979) is augmented with the "slicing method" of Agemi, Kurokawa and Takamura (2000) for lower bounds in the critical case.

math.AP↗
arXiv2017

Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case of non-effective damping. In this paper we extend his result in two aspects by showing that: (I) the blow-up will happen for bigger exponent, which is closely related to the Strauss exponent, the critical number for non-damped semilinear wave equations; (II) such a blow-up result is established for a wider range of the constant than the known non-effective one in the damping term.

math.AP↗
arXiv2016

Global Regularity for Supercritical Nonlinear Dissipative Wave Equations in 3D

The nonlinear wave equation $u_{tt}-Δu +|u_t|^{p-1}u_t=0$ is shown to be globally well-posed in the Sobolev spaces of radially symmetric functions $H^k_{\rm rad}({\bf R}^3)\times H^{k-1}_{\rm rad}({\bf R}^3)$ for all $p\geq 3$ and $k\geq 3$. Moreover, global $C^\infty $ solutions are obtained when the initial data are $C_0^\infty$ and exponent $p$ is an odd integer. The radial symmetry allows a reduction to the one-dimensional case where an important observation of A. Haraux (2009) can be applied, i.e., dissipative nonlinear wave equations contract initial data in $W^{k,q}({\bf R})\times W^{k-1,q}({\bf R})$ for all $k\in[1,2]$ and $q\in [1,\infty]$.

math.AP↗