Blow Up of Solutions to the Second Sound Equation in One Space Dimension

Kato, Keiichi; Sugiyama, Yuusuke

doi:10.2206/kyushujm.67.129

Cited by 7 publications

(15 citation statements)

References 17 publications

(20 reference statements)

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“…In [15], Lindblad has shown that solutions exists globally in time with small initial data. In [8,17], Kato and the author have shown that the equation in (1.1) with c(θ) = 1 + θ and λ = 0, 1 degenerates in finite time, if initial data are smooth, compactly supported and satisfy (1.5) and (1.6). The main theorem of this paper removes the compactness condition on initial data and extends the result in [8,17] to (1.1) with more general c(θ) and 0 ≤ λ < 2.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, applying the method in [17,18] to the equation in (1.1), we can generalize the result in [17,18] to (1.1) with 0 ≤ λ < 2 and c(θ) = 1 + θ. However the compactness condition plays a crucial role in [8,17], since we use the following estimates for bounded solutions under the assumption that initial data are compactly supported:…”

Section: Introductionmentioning

confidence: 99%

“…The second idea for the proof is to divide the situation into the two cases that 2 dxds is bounded, then (1.10) holds and (1.9) can be shown by the use of the Riemann invariant. Hence we can use a variation of the method in [8,17]. The key for the generalization on c(·) is the use of 2 dxds is not bounded, we can use the method in [18].…”

Section: Introductionmentioning

confidence: 99%

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Degeneracy in Finite Time of 1D Quasilinear Wave Equations

Sugiyama¹

2016

SIAM J. Math. Anal.

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We consider the large time behavior of solutions to the following nonlinear wave equation:is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if c(·) 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 in finite time, the equation degenerates. We give a sufficient condition that the equation with 0 ≤ λ < 2 degenerates in finite time. *

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Degeneracy in Finite Time of 1D Quasilinear Wave Equations

Sugiyama¹

2016

SIAM J. Math. Anal.

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“…In [8,17], Kato and the author have shown that the equation in (1) with c(θ) = 1 + θ and λ = 0, 1 degenerates in finite time, if initial data are smooth, compactly supported and satisfy (5) and (6). The main theorem of this paper removes the compactness condition on initial data and extends the result in [8,17] to (1) with more general c(θ) and 0 ≤ λ < 2. In [17,18], the generalization on λ has already been pointed out without a proof.…”

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confidence: 56%

Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

Sugiyama

2017

Evolution Equations &Amp; Control Theory

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We consider the large time behavior of solutions to the following nonlinear wave equation: ∂ 2 t u = c(u) 2 ∂ 2 x u + λc(u)c ′ (u)(∂xu) 2 with the parameter λ ∈ [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(•) 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 so that the equation with 0 ≤ λ < 2 degenerates in finite time.

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“…Speck would like to thank Yuusuke Sugiyama for bringing the references [53,[95][96][97] to his attention, to Michael Dreher for pointing out the references [70,71], to Willie Wong for pointing out the relevance of his work [99], and to an anonymous referee for pointing out the work [65].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

Speck

2017

Analysis & PDE

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Abstract. In three spatial dimension, we study the Cauchy problem for the wave equation −∂ 2 t Ψ + (1 + Ψ) P ∆Ψ = 0 for P ∈ {1, 2}. We exhibit a form of stable Tricomi-type degeneracy formation that has not previously been studied in more than one spatial dimension. Specifically, using only energy methods and ODE techniques, we exhibit an open set of data such that Ψ is initially near 0 while 1 + Ψ vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this phenomenon. The solution remains regular in the following sense: there is a high-order L 2 -type energy, featuring degenerate weights only at the top-order, that remains bounded. When P = 1, we show that any C 1 extension of Ψ to the future of a point where 1 + Ψ = 0 must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the Lorentzian metric corresponding to the wave equation blows up at those points. Thus, our results show that curvature blowup does not always coincide with singularity formation in the solution variable. Similar phenomena occur when P = 2, where the vanishing of 1 + Ψ corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all values of Ψ.The data are compactly supported and are allowed to be large or small as measured by an unweighted Sobolev norm. However, we assume that initially, the spatial derivatives of Ψ are nonlinearly small relative to |∂ t Ψ|, which allows us to treat the equation as a perturbation of the ODE d 2 dt 2 Ψ = 0. We show that for appropriate data, ∂ t Ψ remains quantitatively negative, which simultaneously drives the degeneracy formation and yields a favorable spacetime integral in the energy estimates that is crucial for controlling some top-order error terms. Our result complements those of Alinhac and Lindblad, who showed that if the data are small as measured by a Sobolev norm with radial weights, then the solution is global.

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Blow Up of Solutions to the Second Sound Equation in One Space Dimension

Cited by 7 publications

References 17 publications

Degeneracy in Finite Time of 1D Quasilinear Wave Equations

Degeneracy in Finite Time of 1D Quasilinear Wave Equations

Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

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