The blow-up solutions of the heat equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">F</mml:mi><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>

Ru, Shaolei; Chen, Jiecheng

doi:10.1016/j.jfa.2015.05.005

Cited by 8 publications

(3 citation statements)

References 16 publications

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“…Our method bears also some relation with that of [23]. However, the blowup result in F(L 1 ) of [23] is not put in relation with the size of the data in scale-invariant norms. As such, our blowup result looks more precise, and its proof shorter.…”

Section: Motivations and Overview Of The Main Resultsmentioning

confidence: 98%

“…From the technical point of view, our paper is somehow closer to [17,21], where the authors studied the blowup for different equations, namely diffusion problems with nonlocal quadratic nonlinearity. Our method bears also some relation with that of [23]. However, the blowup result in F(L 1 ) of [23] is not put in relation with the size of the data in scale-invariant norms.…”

Section: Motivations and Overview Of The Main Resultsmentioning

confidence: 98%

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Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

Brandolese

Cortez

2019

Journal of Functional Analysis

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We consider the Cauchy problem of the nonlinear heat equation ut − ∆u = u b , u(0, x) = u0, with b ≥ 2 and b ∈ N. We prove that initial data u0 ∈ S(R n ) (the Schwartz class) arbitrarily small in the scale invariant Besov-normḂ −2/b n(b−1)b/2,q (R n ), can produce solutions that blow up in finite time. The case b = 3 answers a question raised by Yves Meyer. Our result also proves that the smallness assumption put in an earlier work by C. Miao, B. Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.

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Section: Motivations and Overview Of The Main Resultsmentioning

confidence: 98%

Section: Motivations and Overview Of The Main Resultsmentioning

confidence: 98%

Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

Brandolese

Cortez

2019

Journal of Functional Analysis

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“…In [30] author have proved some sharp well-posedness result for (1.1) with β ∈ (1,2] in Sobolev spaces H s (R d ). In [33,10] authors have proved blow up in finite time for (1.1) with β = 2 in some scale invariant Besov space and Fourier-Lebesgue spaces.…”

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confidence: 99%

The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Bhimani¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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The aim of this article is two fold. First, we produce a finite time blow-up solution of the nonlinear fractional heat equation. Secondly, we prove space-time estimates for heat propagator e −tH β associated to fractional harmonic oscillator H β = (−∆ + |x| 2 ) β in modulation spaces M p,p (R d ) (1 ≤ p < ∞). As a consequence, the solution for free fractional heat equation ∂ t u + H β u = 0 grow-up in time near the origin and and decay exponentially in time at infinity.

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Global solution of the 3‐D incompressible Navier‐Stokes equations in the Besov spaces B˙r1,r2,r3σ,1

Ru¹,

Chen²

2017

Math Methods in App Sciences

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In this paper, we construct a more general Besov spacesḂ ,q r 1 ,r 2 ,r 3 and consider the global well-posedness of incompressible Navier-Stokes equations with small data ), which, as far as we know, has not been discussed in other papers. Moreover, the smoothing effect of the solution to Navier-Stokes equations is proved, which may have its own interest. KEYWORDSglobal well-posedness, Navier-Stokes equations, smoothing effect t u + P(u · ∇u) − Δu = 0. It is easy to see that the entries of P(u · ∇u) are first order homogeneous Fourier multipliers applied to bilinear expressions.A huge literature study the well-posedness of the NS equation, for example, Danchin, 3 Frisch et al, 4 and Xin. 5 Kato 6 initiated the study of the NS equations by proving that the NS problem (*) is locally well-posed in L 3 and global well-posed if the initial data are small in L 3 . The method consists in applying a Banach fixed point theorem to the integral formulation of the equation and was generalized by Cannone et al 7 to Besov spaces of negative index of regularity. More precisely, they proved that if the initial data are small in the Besov spaceḂ −1+ N p p,∞ (for p < ∞), then there is a unique, global in time solution. The study 6790

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The blow-up solutions of the heat equations in FL1(RN)

Cited by 8 publications

References 16 publications

Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Global solution of the 3‐D incompressible Navier‐Stokes equations in the Besov spaces B˙r1,r2,r3σ,1

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