Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

The pyrolysis of hexachloroethane

The thermal decomposition of hexachloroethane in the presence of chlorine has been studied over the temperature range 340-400°C and a pressure range of 0.5°2.5 atm. The rate of the unimolecular C? C bond spliting reaction can be described by the Arrhenius equation Comparison of these rate data with thermodynamic data suggests a combination rate for CCl₃ radicals which is consistent with earlier measurements.     

Biphenyltetrols and Dibenzofuranones from Oxidative Coupling of Resorcinols with 4-Alkylpyrocatechols: New Ciues to the Mechanism of Insect Cuticle Sclerotization

Oxidation of 4-alkylpyrocatechols 2 by means of an insect diphenoloxidase (laccase) or K₃ [Fe(CN)₆] yields, in the presence of resorcinols 1 (R² ? H), complex mixtures of products from which biphenyltetrols 3 (R³ ? H) and dibenzofuranones 5 and 6 were isolated. It is suggested that similar homo-coupling products are formed from pyrocatechols 2b and 2c in insects during cuticle sclerotization.     

Rotating sector study of the gas phase photolysis of the carbon tetrachloride–cyclohexane system

The rate constant for the combination of trichloromethyl radicals in the gas phase has been measured by applying the rotating sector technique to the gas phase carbon tetrachloride–cyclohexane photochemical system. A temperature-independent rate constant, k₅, of 3.9 ± 1.0 × 10¹² cc mole^?1 sec^?1 was found. Arrhenius parameters for the reaction were found to be given by the expression log k₄ = 11.79 – (10,700/2.3 RT).     

Homogeneous photochemical oxidation via singlet O2 in supercritical CO2

This communication describes the reaction chemistry of singlet oxygen in supercritical carbon dioxide, demonstrating rapid and quantitative conversion of alpha-terpinene to ascaridole.     

Synthesis and structures of crystalline Li, Al and Sn(II) 1-azaallyls and beta-diketiminates derived from [Li{mu,eta3-N(SiMe3)C(Ad)C(H)SiMe3}]2 (Ad = 1-adamantyl)

The crystalline dimeric 1-azaallyllithium complex [Li{mu,eta(3-N(SiMe3)C(Ad)C(H)SiMe3}]2 (1) was prepared from equivalent portions of Li[CH(SiMe3)2] and 1-cyanoadamantane (AdCN). Complex was used as precursor to each of the crystalline complexes 2-8 which were obtained in good yield. By 1-azaallyl ligand transfer, 1 afforded (i) [Al{eta3-N(SiMe3)C(Ad)C(H)SiMe3}{kappa1-N(SiMe3)C(Ad)=C(H)SiMe3-E}Me] (5) with [AlCl2Me](2), (ii) [Sn{eta3-N(SiMe3)C(Ad)C(H)SiMe3}2] (7) with Sn[N(SiMe3)2]2, and (iii) [Li(N{C(Ad)=C(H)SiMe3-E}{Si(NN)SiMe3})(thf)2] (8) with the silylene Si[(NCH(2)Bu(t))2C6H(4)-1,2] [= Si(NN)]. By insertion into the C[triple bond, length as m-dash]N bond of the appropriate cyanoarene RCN, gave the beta-diketiminate [Li{mu-N(SiMe3)C(Ad)C(H)C(R)NSiMe3}]2 [R = Ph (2), C(6)H(4)Me-4 (3)], and yielded [Al{kappa2-N(SiMe3)C(Ad)C(H)C(Ph)NSiMe3}{kappa1-N(SiMe3)C(Ad)=C(H)SiMe3-E}Me] (6). The beta-diketiminate [Al{kappa2-N(SiMe3)C(Ad)C(H)C(Ph)NSiMe3}Me2] (4) was prepared from 2 and [AlClMe2]2. The X-ray structures of 1 and 3-8 are presented. Multinuclear NMR spectra in C6D6 or C6D5CD3 have been recorded for each of 1-8; such data on 8 revealed that in solution two minor isomers were also present.