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Takahisa Inui

Publications and source records attributed to Takahisa Inui.

33 records · Page 2Linked to original sources
arXiv2020

Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data

In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time $t=0$. Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtain the scattering result and its asymptotic order for the data in the energy space $H^1\times L^2$ when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on $r$ when the initial data is in $L^r$ ($1\leq r < 2$). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for $L^r$-data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

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arXiv2019

Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph

We consider the Schrödinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, $δ$, or $δ'$ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is based on the argument by Murphy and Nakanishi, who treated the long-range nonlinear Schrödinger equation with a general potential in the Euclidean space, in order to consider general boundary conditions.

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arXiv2019

Endpoint Strichartz estimate for the damped wave equation and its application

Recently, the Strichartz estimates for the damped wave equation was obtained by the first author except for the wave endpoint case. In the present paper, we give the Strichartz estimate in the wave endpoint case. We slightly modify the argument of Keel--Tao. Moreover, we apply the endpoint Strichartz estimate to the unconditional uniqueness for the energy critical nonlinear damped wave equation. This problem seems not to be solvable as the perturbation of the wave equation.

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arXiv2019

Scattering for the quadratic nonlinear Schrödinger system in $\mathbb{R}^5$ without mass-resonance condition

We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $κ>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $κ= 1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $κ\neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Kenig--Merle. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.

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arXiv2019

The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

For the linear damped wave equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear damped wave equation (NLDW) $\partial_t^2 u - Δu +\partial_t u = |u|^{\frac{4}{d-2}}u$, $(t,x) \in [0,T) \times \mathbb{R}^d$, where $3 \leq d \leq 5$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.

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arXiv2019

Scattering for the one-dimensional Klein-Gordon equation with exponential nonlinearity

We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. We prove that any energy solution has a global bound of the $L^6_{t,x}$ space-time norm, and hence scatters in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$ as $t\rightarrow\pm \infty$. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364 (2012), no. 3, 1571--1631). However, since well-posedness in $H^{1/2}(\mathbb{R}) \times H^{-1/2}(\mathbb{R})$ for NLKG with the exponential nonlinearity holds only for small initial data, we use the $L_t^6 W^{s-1/2,6}_x$-norm for some $s>\frac{1}{2}$ instead of the $L_{t,x}^6$-norm, where $W_x^{s,p}$ denotes the $s$-th order $L^p$-based Sobolev space.

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arXiv2018

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

We consider a mass-critical system of nonlinear Schödinger equations \begin{align*} \begin{cases} i\partial_t u +Δu =\bar{u}v,\\ i\partial_t v +κΔv =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} where $(u,v)$ is a $\mathbb{C}^2$-valued unknown function and $κ>0$ is a constant. If $κ=1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(ϕ,ψ)$, where $M(u,v)$ denotes the mass and $(ϕ,ψ)$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson \cite{MR3406535}. Scattering is also obtained without mass-resonance condition under the restriction that $(u,v)$ is radially symmetric.

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arXiv2018

Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $κ>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $κ=1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $κ\neq 1/2$ and we show blow-up or grow-up when $d=4$.

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arXiv2018

$L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - Δu + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^β) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $β= (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.

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arXiv2017

The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping

We consider the following semilinear wave equation with time-dependent damping. \begin{align} \tag{NLDW} \left\{ \begin{array}{ll} \partial_t^2 u - Δu + b(t)\partial_t u = |u|^{p}, & (t,x) \in [0,T) \times \mathbb{R}^n, \\ u(0,x)=\varepsilon u_0(x), u_t(0,x)=\varepsilon u_1(x), & x \in \mathbb{R}^n, \end{array} \right. \end{align} where $n \in \mathbb{N}$, $p>1$, $\varepsilon>0$, and $b(t)\thickapprox (t+1)^{-β}$ with $β\in [-1,1)$. It is known that small data blow-up occurs when $1 p_F$, where $p_F:=1+2/n$ is the Fujita exponent. The sharp estimate of the lifespan was well studied when $1<p< p_F$. In the critical case $p=p_F$, the lower estimate of the lifespan was also investigated. Recently, Lai and Zhou obtained the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)=1$. In the present paper, we give the sharp upper estimate of the lifespan when $p=p_F$ and $b(t)\thickapprox (t+1)^{-β}$ with $β\in [-1,1)$ by the Lai--Zhou method.

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arXiv2017

A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if $\left\Vert u_0 \right\Vert_{L^2}^2<4π$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schrödinger equation", Analysis & PDE 8 (2015), no. 5, 1101--1112. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_0$ satisfies that $\left\Vert u_0 \right\Vert_{L^2}^2=4π$ and the momentum $P(u_0)$ is negative.

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arXiv2016

Global Dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential

We consider the focusing mass supercritical semilinear Schrödinger equation with a repulsive Dirac delta potential on the real line (deltaNLS). Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave of NLS without a potential to determine the global behavior of the solution. Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave and smaller than that of the standing wave of deltaNLS.

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arXiv2016

The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.

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