Nondegeneracy of blowup for semilinear heat equations

Giga, Yoshikazu; Kohn, Robert V.

doi:10.1002/cpa.3160420607

Cited by 276 publications

(326 citation statements)

References 28 publications

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“…and generalize to this equation the results first proved in the real case by Giga and Kohn [GK85,GK87,GK89]. Our equation (1) appears then as a "twin" of equation (7).…”

Section: Introductionsupporting

confidence: 66%

“…Proof: See Theorem 2.1 page 850 in [GK89]. Note that the proof of Giga and Kohn is valid also when u is complex valued.…”

Section: Asymptotic Behavior Of U(t)mentioning

confidence: 95%

“…Note also that the stability of such a profile with respect to initial data has been proved by Fermanian Kammerer, Merle and Zaag in [FKZ00] and [FKMZ00]. For more results on equation (3), see [Bal77], [GK85], [GK87], [GK89], [HV93], [HV94], [MM04], [MM09], [MZ98], [MZ00], [Miz07] and [QS07].…”

Section: Introductionmentioning

confidence: 93%

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Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation

Nouaili

Zaag

2015

Communications in Partial Differential Equations

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We construct a solution to a complex nonlinear heat 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. We note that the real and imaginary parts of the constructed solution blow up simultaneously, with the imaginary part dominated by the real.Mathematical Subject classification: 35K57, 35K40, 35B44.

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“…and generalize to this equation the results first proved in the real case by Giga and Kohn [GK85,GK87,GK89]. Our equation (1) appears then as a "twin" of equation (7).…”

Section: Introductionsupporting

confidence: 66%

“…Proof: See Theorem 2.1 page 850 in [GK89]. Note that the proof of Giga and Kohn is valid also when u is complex valued.…”

Section: Asymptotic Behavior Of U(t)mentioning

confidence: 95%

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Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation

Nouaili

Zaag

2015

Communications in Partial Differential Equations

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“…On the one hand, the variational characterization of Q; i.e. : for v ∈ H 1 (R), if 0 < v 2 ≤ Q 2 and E(v) ≤ 0, (5) then there exists λ 0 > 0, x 0 ∈ R / v = λ 1/2 0 Q(λ 0 (x − x 0 )), which provides the following Gagliardo-Nirenberg inequality with best constant (see Weinstein [17]): (6) for all v ∈ H 1 (R), 1 6…”

mentioning

confidence: 99%

“…Indeed, in the case of the nonlinear heat equations u t = ∆u+u p in R N , with some restriction on p, the blow-up rate and profile (and their stability) have been determined; see for example Giga and Kohn [5] and FermanianKammerer, Merle and Zaag [4].…”

mentioning

confidence: 99%

Stability of Blow-Up Profile and Lower Bounds for Blow-Up Rate for the Critical Generalized KdV Equation

Martel¹,

Merle²

2002

The Annals of Mathematics

117

156

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The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence).From [15], there do exist for this equation solutions u(t) such that |u(t)| H 1 → +∞ as t ↑ T , where T ≤ +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs.For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T . From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.

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