Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Mizoguchi, Noriko; Souplet, Philippe

doi:10.1016/j.aim.2019.106763

Cited by 17 publications

(10 citation statements)

References 40 publications

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“…Observe that of course f α ∈ L p for every p (if α = 0), thus Theorem 1.2 reveals that we can control initial data beyond L p (there is an enormous literature on nonlinear heat equations with Cauchy data in L p ; see e.g. [26,38] and the references therein). (iii) The results in Theorem 1.2 look interesting because they are in sharp contrast to what happens for the standard heat equation with the above nonlinearity and λ = 1 (focusing case), where for real constant initial data (hence in M ∞,1 ), even small, one has always blow-up in finite time for every k = 0, as one sees at once by solving the ordinary differential equation u t = u k+1 , u real (again the literature in this connection is large; see [38] for a comprehensive survey).…”

Section: Introduction and Discussion Of The Resultsmentioning

confidence: 99%

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Bhimani¹,

Manna²,

Nicola³

et al. 2020

Preprint

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We study the Hermite operator H = −∆ + |x| 2 in R d and its fractional powers H β , β > 0 in phase space. Namely, we represent functions f via the socalled short-time Fourier, alias Fourier-Wigner or Bargmann transform V g f (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of V g f , that is in terms of membership to modulation spaces M p,q , 0 < p, q ≤ ∞. We prove the complete range of fixed-time estimates for the semigroup e −tH β when acting on M p,q , for every 0 < p, q ≤ ∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing.As an application, we establish global well-posedness for the nonlinear heat equation for H β with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e −ct as the solution of the corresponding linear equation, where c = d β is the bottom of the spectrum of H β . This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data -hence in M ∞,1 .

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Section: Introduction and Discussion Of The Resultsmentioning

confidence: 99%

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Bhimani¹,

Manna²,

Nicola³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In what follows, we set r = |y|. Due to (35) and (36), it is natural to construct a solution of the form:…”

Section: Preliminarymentioning

confidence: 99%

“…Remark 1.3. A recent result [36] shows that the critical L q norm blow-up does occur for possibly non-radial solutions of (1) if the blow-up is of type I. The solution u as in Theorem 1.1 exhibits type II blow-up.…”

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confidence: 95%

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Mukai¹,

Seki²

2021

DCDS

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We are concerned with blow-up mechanisms in a semilinear heat equation:ut = ∆u + |x| 2a u p , x ∈ R N , t > 0, where p > 1 and a > −1 are constants. As for the Fujita equation, which corresponds to a = 0, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if N ≥ 11 andwhere T is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case a = 0. As a consequence, we obtain an example of solutions that blow up at x = 0, the zero point of potential |x| 2a with a > 0, and behave in non-self-similar manner for N > 10 + 8a. This last result is in contrast to backward self-similar solutions previously obtained for N < 10 + 8a, which blow up at x = 0.

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“…So, m = 1+2/d is a critical power for the global solutions of (3) with non-negative initial data. The blowup behavior of the solutions of (3) were studied in [20,33,35,46,47] (see also [42]). It seems that the sign-change solutions are more complicated.…”

Section: Introductionmentioning

confidence: 99%

Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$

Chen¹,

Wang²,

Wang³

2022

Preprint

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We consider the Cauchy problem for the complex valued semi-linear heat equationwhere m ≥ 2 is an integer and the initial data belong to super-critical spaces E s σ for which the norms are defined byWe obtain the global existence and uniqueness of the solutions if the initial data belong to E s σ (s < 0, σ ≥ d/2 − 2/(m − 1)) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in E s σ are not required for the global solutions. Moreover, we show that the error between the solution u and the iteration solution u (j) is C j /(j !) 2 . Similar results also hold if the nonlinearity u m is replaced by an exponential function e u − 1. 2020 MSC: 35K58.

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Optimal condition for blow-up of the critical L norm for the semilinear heat equation

Cited by 17 publications

References 40 publications

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$

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