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Yuusuke Sugiyama

Publications and source records attributed to Yuusuke Sugiyama.

16 recordsLinked to original sources
arXiv2026

Loss of regularity for solutions to 1D degenerate quasilinear wave equations

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation \( u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x \) with \(c(t,x)\sim x^p\), and then the quasilinear equation \( u_{tt}=(u^{2a}u_x)_x. \) The initial data are assumed to satisfy \( u_0(x)\sim x^\alpha \) and \( u_1(x)\sim x^\beta \) near the degenerate point \(x=0\). In the linear problem, \(p\) describes the strength of the degeneracy, while \(\alpha\) and \(\beta\) describe the regularity of the initial data. In the quasilinear problem, \(\alpha\) also determines the initial degeneracy through \(u_0(x)^a\sim x^{a\alpha}\). Our main concern is the actual occurrence of loss of regularity, namely, the phenomenon in which the regularity of \(u(t,\cdot)\) near \(x=0\) becomes lower than that of \(u_0\) for \(t>0\). Previously, such a loss had mainly been established for special linear equations with coefficients depending only on time. In our previous work on the quasilinear equation, we proved local well-posedness without loss of regularity when \(\beta\geq\alpha\). In the present paper, we show that this condition is also necessary. More precisely, if \(\beta<\alpha\), then \( C_1tx^\beta\leq u(t,x)\leq C_2x^\beta \) near \(x=0\) for sufficiently small positive time. Thus the order of the solution changes from that of \(u_0\) to that of \(u_1\), and an actual loss of regularity occurs for both linear equations with time-space dependent coefficients and degenerate quasilinear equations.

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arXiv2026

Formation of singularities for a family of one-dimensional quasilinear wave equations beyond the variational case

We consider finite-time singularity formation for classical solutions to the following parameterized nonlinear wave equation: \[ u_{tt}=c(u)^2u_{xx}+\lambda c(u)c'(u)(u_x)^2, \] where \(\lambda\in[0,2]\) is a parameter. In previous works, it is known that finite-time blow-up solutions exist for \(0<\lambda\le1\) and for \(\lambda=2\). The cases \(\lambda=1\) and \(\lambda=2\) are known as the variational wave equation and the \(p\)-system, respectively, and they possess symmetric structures or conservation laws. For \(0<\lambda<1\), the blow-up construction is based on the fact that, after decomposing the wave into two Riemann variables, one component can be kept sufficiently small and hence does not prevent the other component from blowing up. In the present paper, we treat the remaining intermediate case \(1<\lambda<2\). This case is essentially different from the previous one, since both Riemann variables may grow. The key idea is to introduce suitable supremum functions for the Riemann variables and to derive a comparison principle through right Dini derivatives. This allows us to control the relative size of the two components and to obtain a Riccati-type differential inequality for the dominant supremum function.

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arXiv2026

Lifespan of Classical Solutions to One-Dimensional Quasilinear Wave Equations

In this paper, we consider the upper and lower bounds of the lifespan of classical solutions of the Cauchy problem for the one-dimensional quasilinear wave equation $u_{tt}-c(u_x)^2u_{xx}=0$ where the derivative of $c(\theta)$ tends to $0$ near the origin. In particular, our result shows that the lifespan of the solution extends algebraically depending on the smallness of the initial data. Furthermore, we also show that when $c(\theta)$ is flat at the origin ($c'(\theta)$ and any higher order derivatives vanish at the origin), the lifespan extends exponentially depending on the smallness of the initial data. Our proof is based on the method of Lax's characteristics and Riemann invariants.

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arXiv2026

Large data global well-posedness for a one-dimensional quasilinear wave equation

In this paper, we prove global well-posedness with large initial data for the one-dimensional quasilinear wave equation $$ u_{tt}=c(u)^2u_{xx}, \qquad (t,x)\in (0,T)\times\R, $$ where \(c\) is a positive, bounded, monotonically increasing function with bounded derivative. This result gives a partial resolution of an open problem posed by Glassey, Hunter and Zheng on the global existence of smooth solutions to this equation for large initial data. Our proof is based on upper and lower estimates for the Riemann variables via a new comparison principle.

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arXiv2026

Wave front set of solutions to the fractional Schr\"{o}dinger equation

In this paper, we characterize the wave front sets of solutions to fractional Schr\"{o}dinger equations \(i\partial_{t}u =(-\Delta)^{\theta/2}u + V(x)u\) with $0<\theta <2$ via the wave packet transform (short-time Fourier transform). We clarify the relationship between the order \(\theta\) of the fractional Laplacian and the growth rate of the potential in the problem of propagation of singularities. In particular, we present a theorem that bridges the propagation mechanisms of singularities for the Schr\"odinger and wave equations.

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arXiv2023

Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or $0$, the solution exists globally in time or blows up in finite time, respectively. We prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient $a$ satisfies the following condition $$ |a(t,x)- μ_0| \leq a_1(t) + a_2 (x), $$ where $μ_0 \geq 0$ and $a_1$ and $a_2$ are integrable functions with $t$ and $x$. Under this condition, we show the global existence and the blow-up with small initial data, when $μ_0 >0$ and $μ=0$ respectively.

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arXiv2022

Local solvability for a quasilinear wave equation with the far field degeneracy: 1D case

We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristic and the contraction mapping principle via weighted $L^\infty$ estimates.

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arXiv2022

Formation of singularities for a family of 1D quasilinear wave equations

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + λc(u)c'(u)( u_x)^2$ with the real parameter $λ$. In previous works, it was reported that there exist finite time blow-up solutions with $λ=1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $λ=1$ to the case with $λ\in (0,1]$ by using a new $L^{2/λ}$ estimate. Moreover, some properties for the blow-up solution including the Hölder continuity are also discussed.

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arXiv2019

Remark on global existence of solutions to the 1D compressible Euler equation with time-dependent damping

In this paper, we consider the 1D compressible Euler equation with the damping coefficient $λ/(1+t)^μ$. Under the assumption that $0\leq μ<1$ and $λ>0$ or $μ=1$ and $λ> 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results.

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arXiv2018

Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation

The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.

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arXiv2017

Existence and nonexistence theorems for global weak solutions to quasilinear wave equations for the elasticity

In this paper, by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor \cite{LPT,LPS}, we prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations $v_{tt} =c (|v|^{s-1} v)_{xx}$ with the nonnegative initial data $v_{0}(x)$ and $ s > 1$. This result is an extension of the results in the second author's paper \cite{Su}, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on $\int_{-\infty}^{\infty} v_{1}(x) dx$ and the non-degeneracy condition $ v_{0}(x) \geq c_{0} > 0$ on the initial data.

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arXiv2017

Singularity formation for the 1D compressible Euler equation with variable damping coefficient

In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial data are small perturbations of constant states.

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arXiv2017

Degeneracy in finite time of 1D quasilinear wave equations II

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + λc(u)c'(u)(\partial_x u)^2$ with the parameter $λ\in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq λ< 2$ degenerates in finite time.

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arXiv2015

Asymptotic expansion of solutions to the drift-diffusion equation with fractional dissipation

The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian. When the exponent of the fractional Laplacian is large, large-time behavior of solutions is known. However, when the exponent is small, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with small exponent is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.

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arXiv2013

Global existence and blow-up of solutions to some quasilinear wave equation in one space dimension

We consider the global existence and blow up of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = \partial_x(c(u)^2 \partial_x u)$, which has richly physical backgrounds. Under the assumption that $c(u(0,x))\geq δ$ for some $δ>0$, we give sufficient conditions for the existence of global smooth solutions and the occurrence of two types of blow-up respectively. One of the two types is that $L^{\infty}$-norm of $\partial_t u$ or $\partial_x u$ goes up to the infinity. The other type is that $c(u)$ vanishes, that is, the equation degenerates.

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