Erratum: long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term - III

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Abstract

Estimate (4.5) of proposition 2 of Milani (1991) is shown to be wrong, since it is not independent of ε as ε ↓ 0 as claimed. The correct estimate would be |▽ψ(t)| ≤ C(1 + εt)-p{∥ψ0s(1) + ∥ψ1s(0)}, but this estimate is useless, since the uniform bound on the function (1 + εt)p0 t (1 + ε(t - θ))-p(1 + εθ)-p dθ, which we would then have to consider in (4.12), is now of order 1/ε. This invalidates the proof of the main lemma, and hence of theorem 4.

Original languageEnglish
Pages (from-to)367-368
Number of pages2
JournalNonlinear Analysis, Theory, Methods and Applications
Volume27
Issue number3
DOIs
Publication statusPublished - Jan 1 1996

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Quasilinear Hyperbolic Equation
Singular Perturbation
Small Parameter
Dissipation
Term
Estimate
Uniform Bound
Proposition
Lemma
Theorem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Erratum: long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term - III",
abstract = "Estimate (4.5) of proposition 2 of Milani (1991) is shown to be wrong, since it is not independent of ε as ε ↓ 0 as claimed. The correct estimate would be |▽ψ(t)| ≤ C(1 + εt)-p{∥ψ0∥s(1) + ∥ψ1∥s(0)}, but this estimate is useless, since the uniform bound on the function (1 + εt)p ∫0 t (1 + ε(t - θ))-p(1 + εθ)-p dθ, which we would then have to consider in (4.12), is now of order 1/ε. This invalidates the proof of the main lemma, and hence of theorem 4.",
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N2 - Estimate (4.5) of proposition 2 of Milani (1991) is shown to be wrong, since it is not independent of ε as ε ↓ 0 as claimed. The correct estimate would be |▽ψ(t)| ≤ C(1 + εt)-p{∥ψ0∥s(1) + ∥ψ1∥s(0)}, but this estimate is useless, since the uniform bound on the function (1 + εt)p ∫0 t (1 + ε(t - θ))-p(1 + εθ)-p dθ, which we would then have to consider in (4.12), is now of order 1/ε. This invalidates the proof of the main lemma, and hence of theorem 4.

AB - Estimate (4.5) of proposition 2 of Milani (1991) is shown to be wrong, since it is not independent of ε as ε ↓ 0 as claimed. The correct estimate would be |▽ψ(t)| ≤ C(1 + εt)-p{∥ψ0∥s(1) + ∥ψ1∥s(0)}, but this estimate is useless, since the uniform bound on the function (1 + εt)p ∫0 t (1 + ε(t - θ))-p(1 + εθ)-p dθ, which we would then have to consider in (4.12), is now of order 1/ε. This invalidates the proof of the main lemma, and hence of theorem 4.

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