Well posedness in the C∞ class for utt=a(u)Δu

Manfrin, Renato

doi:10.1016/s0362-546x(97)00703-7

Cited by 7 publications

(11 citation statements)

References 16 publications

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“…Manfrin (e.g. [13,14]) showed the existence and the uniqueness of solutions of the Cauchy problem of some quasilinear hyperbolic equations including ∂ 2 t u = u 2 u and…”

Section: From (319) U(t X) Is a Monotone Decreasing Function Of T mentioning

confidence: 99%

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

Kato¹,

Sugiyama²

2013

Kyushu J. Math.

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Abstract. In this paper, we study blow ups of solutions to the second sound equation, which is more natural than the second sound equation in Landau-Lifshitz's text in large time. We assume that the initial data satisfies u(0, x) ≥ δ > 0 for some δ. We give sufficient conditions that two types of blow up occur: one of the two types is that L ∞ -norm of ∂ t u or ∂ x u goes up to the infinity; the other type is that u vanishes, that is, the equation degenerates.

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“…Manfrin (e.g. [13,14]) showed the existence and the uniqueness of solutions of the Cauchy problem of some quasilinear hyperbolic equations including ∂ 2 t u = u 2 u and…”

Section: From (319) U(t X) Is a Monotone Decreasing Function Of T mentioning

confidence: 99%

“…is not treated in [13,14]. Here we do not discuss the existence and the uniqueness of solutions of (1.1) with u(0, x), ∂ t u(0, x) ∈ C ∞ 0 (R).…”

Section: From (319) U(t X) Is a Monotone Decreasing Function Of T mentioning

confidence: 99%

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

Kato¹,

Sugiyama²

2013

Kyushu J. Math.

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“…Similar types of degenerate behavior occur in other PDE contexts: gradient flows such as the porous medium equation or the parabolic p-Laplacian flow (see for example the monographs [20,57]); higher order diffusion such as the thin film equation [23,25,[35][36][37]; weakly hyperbolic equations [19,40], in particular the compressible Euler equations near vacuum [17,30]. Indeed, many of the techniques in this paper owe inspiration to previous work on degenerate parabolic and hyperbolic equations.…”

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confidence: 56%

“…A key difficulty of working in the original frame is that the degeneracy at the endpoints will be time-dependent. In order to remove this time-dependence we switch to a moving frame, an approach that is common in degenerate hyperbolic and parabolic equations (see for example [19,23,25,38,40]). Recalling the hydrodynamic formulation (1.10), we let X be the Lagrangian map associated to the vector field b; in other words…”

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confidence: 99%

Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

Germain

Harrop-Griffiths

Marzuola

2019

Comm Pure Appl Math

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We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data. © 2019 Wiley Periodicals, Inc.

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“…Manfrin in [17] have established the local existence and the uniqueness for following 1D degenerate quasilinear wave equations with u 0 , u 1 ∈ C ∞ 0 (R n ):…”

Section: Known Resultsmentioning

confidence: 99%

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

Sugiyama¹

2022

Preprint

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We study the Cauchy problem for the quasilinear wave equationx with a ≥ 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) ≥ 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) → 0 as |x| → ∞. Furthermore, to prove the local wellposedness, 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 ∞ estimates.

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Well posedness in the C∞ class for utt=a(u)Δu

Cited by 7 publications

References 16 publications

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

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

Existence and Uniqueness of Solutions for a Quasilinear KdV Equation with Degenerate Dispersion

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

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