Searcharxiv⌕ Search

arXiv subjects

Masahiro Ikeda

Publications and source records attributed to Masahiro Ikeda.

107 records · Page 6Linked to original sources
arXiv2018

Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method

This paper is concerned with the blowup phenomena for initial-boundary value problem for certain semi linear parabolic, dispersive and hyperbolic equations in cone-like domain. The result proposes a unified treatment of estimates for lifespan of solutions to the problem by test function method. The Fujita exponent p=1 + 2/N appears as a threshold of blowup phenomena for small data when $C_{Σ}=R^N$ , but the case of cone-like domain with boundary the threshold changes and explicitly given via the first eigenvalue of corresponding Laplace-Beltrami operator with Dirichlet boundary condition as in Levine-Meier in 1989.

math.AP↗
arXiv2018

Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.

math.AP↗
arXiv2017

Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain

In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial data $(f,tau g)in H^1_0(Omega)times L^2(Omega)$ satisfying $(f+tau g)log |x|in L^1(Omega)$ with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to the problem is given as the following {it double exponential type} when $p=2$: [ lifespan(u) leq exp[exp(Cep^{-1})] . ] The crucial idea is to use test functions which approximates the harmonic function $log |x|$ satisfying Dirichlet boundary condition and the technique for derivation of lifespan estimate in Ikeda-Sobajima(arXiv:1710.06780).

math.AP↗
arXiv2017

Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical space-dependent damping term (DW:$V$). The main result of the present paper is to give a solution of the problem and to provide a sharp estimate for lifespan for such a solution when $\frac{N}{N-1}<p\leq p_S(N+V_0)$, where $p_S(N)$ is the Strauss exponent for (DW:$0$). The main idea of the proof is due to the technique of test functions for (DW:$0$) originated by Zhou--Han (2014, MR3169791). Moreover, we find a new threshold value $V_0=\frac{(N-1)^2}{N+1}$ for the coefficient of critical and singular damping $|x|^{-1}$.

math.AP↗
arXiv2017

Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+μ)$, where $p_F(N)$ denotes the Fujita exponent for the nonlinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonlinear wave equation in $n$-dimension with $μ=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+μ)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the conjugate linear equation $\pa_t^2Φ-ΔΦ-\pa_t(\fracμ{1+t}Φ)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.

math.AP↗
arXiv2017

Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations

Blow-up phenomena ofvweakly coupled systems of several evolution equations, especially complex Ginzburg-Landau equationsvis shown by a straightforward ODE approach not so-called test-function method, which gives the natural blow-up rate. The difficulty of the proof is that, unlike the single case, terms which come from the fact that the Laplacian cannot be absorbed into the weakly coupled nonlinearities. A similar ODE approach is applied to heat systems by Mochizuki to obtain the lower estimate of lifespan.

math.AP↗
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.

math.AP↗
arXiv2017

Software Defined Media: Virtualization of Audio-Visual Services

Internet-native audio-visual services are witnessing rapid development. Among these services, object-based audio-visual services are gaining importance. In 2014, we established the Software Defined Media (SDM) consortium to target new research areas and markets involving object-based digital media and Internet-by-design audio-visual environments. In this paper, we introduce the SDM architecture that virtualizes networked audio-visual services along with the development of smart buildings and smart cities using Internet of Things (IoT) devices and smart building facilities. Moreover, we design the SDM architecture as a layered architecture to promote the development of innovative applications on the basis of rapid advancements in software-defined networking (SDN). Then, we implement a prototype system based on the architecture, present the system at an exhibition, and provide it as an SDM API to application developers at hackathons. Various types of applications are developed using the API at these events. An evaluation of SDM API access shows that the prototype SDM platform effectively provides 3D audio reproducibility and interactiveness for SDM applications.

cs.MM↗
arXiv2016

Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.

math.AP↗
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.

math.AP↗
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.

math.AP↗
arXiv2013

Null structure in a system of quadratic derivative nonlinear Schrödinger equations

We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.

math.AP↗
arXiv2012

Instabilities and turbulence-like dynamics in an oppositely driven binary particle mixture

Using extensive particle-based simulations, we investigate out-of-equilibrium pattern dynamics in an oppositely driven binary particle system in two dimensions. A surprisingly rich dynamical behavior including lane formation, jamming, oscillation and turbulence-like dynamics is found. The ratio of two friction coefficients is a key parameter governing the stability of lane formation. When the friction coefficient transverse to the external force direction is sufficiently small compared to the longitudinal one, the lane structure becomes unstable to shear-induced disturbances, and the system eventually exhibits a dynamical transition into a novel turbulence-like phase characterized by random convective flows. We numerically construct an out-of-equilibrium phase diagram. Statistical analysis of complex spatio-temporal dynamics of the fully nonlinear turbulence-like phase suggests its apparent reminiscence to the swarming dynamics in certain active matter systems.

cond-mat.soft↗
arXiv2011

A remark on the algebraic normal form method applied to the Dirac-Klein-Gordon system in two space dimensions

We consider the massive Dirac-Klein-Gordon system in two space dimensions. Under the non-resonace mass condition, we show that the solution is asymptotically free if the initial data are sufficiently small in a suitable weighted Sobolev space. In particular, it turns out that the Dirac component of the DKG system tends to a solution of the free Dirac equation. Our proof is based on the algebraic normal form method.

math.AP↗