A Simplified Proof of a Liouville Theorem for Nonnegative Solution of a Subcritical Semilinear Heat Equations

Nouaili, Nejla

doi:10.1007/s10884-008-9121-6

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Introduction1

Proof Of Proposition 461

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2010

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Cited by 2 publications

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“…This result has already been proved by Merle and Zaag [MZ98a] and [MZ00] (see also Nouaili [Nou08]) when δ = 0. In that case, M (0) = +∞, which means that any L ∞ entire solution of (1.10), with no restriction on the size of its norm, is trivial (i.e.…”

Section: Introductionsupporting

confidence: 70%

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up

Nouaili¹,

Zaag²

2010

Trans. Amer. Math. Soc.

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Section: Introductionsupporting

confidence: 70%

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up

Nouaili¹,

Zaag²

2010

Trans. Amer. Math. Soc.

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“…We just have to project Eqs. (17) and (21) to get equations satisfied by the different coordinates of the decomposition (42). More precisely, the proof will be carried out in 3 subsections, -In the first subsection, we deal with Eq.…”

Section: Proof Of Proposition 46mentioning

confidence: 99%

Blow-up profile for the complex Ginzburg–Landau equation

Masmoudi

Zaag

2008

Journal of Functional Analysis

122

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We construct a solution to the complex Ginzburg-Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data.

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A Simplified Proof of a Liouville Theorem for Nonnegative Solution of a Subcritical Semilinear Heat Equations

Abstract: We give a new proof of the Liouville theorem proved by Merle and Zaag for nonnegative solutions of the semilinear heat equation with power nonlinearity. Our proof has a pedagogical interest and is based on Kaplan's blow-up criterion.

Cited by 2 publications

References 12 publications

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up

A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up

Blow-up profile for the complex Ginzburg–Landau equation

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