A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

LetDR ^N be a region with smooth boundaryD. Letp·q>1,p, q1. We consider the system:u _t=u+v ^p,v _t=u+u ^q inD×[0, ) withu=v=0 inD×[0, ) andu ₀,v ₀ nonnegative. Let=max(p, q). We show that ifD isR ^N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (+1)/(pq–1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (+1)/(pq–1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R ^N, (a) holds with equality. An explicit formula forpc(D) is given.This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.     

Solution properties of cellulose triacetate. I. Fractional precipitation

Using petroleum ether as a precipitant and chloroform–acetone mixture as solvent, six fractions of cellulose triacetate were obtained by fractional precipitation. The molecular weight of each fraction was obtained from osmotic pressure measurements carried out on chloroform solutions. It was shown that virtual nonfractionability of cellulose triacetate with regard to molecular weight occurred in this system. This behavior was also observed in attempts to fractionate it from tetrachloroethane or acetic acid solutions. These results are explained by the hypothesis that specific polymer–solvent interaction takes place due to hydrogen bonding.     

Incoherent inelastic neutron scattering studies of potassium bicarbonate and caesium hydrogen dinitrate

Inelastic neutron scattering spectra of KHCO₃ and CsH(NO₃)₂ have been obtained in the region 400 → 2400 and 400 → 2800 cm^?1 respectively. The in- and out-of-phase bending vibrations of the hydrogen bonds have been observed and assigned. For CsH(NO₃)₂ the two bending modes are closer in frequency than in KHCO₃ and they are not resolved from the antisymmetric stretch.     

The synthesis and structural study of five-coordinate triorganotin(IV) tropolonates and dibenzoylmethanates

The complexes [1,3-diphenyl-1,3-propanedionato]tricyclohexyltin(IV), (tropolonato)triphenyltin(IV), and (tropolonato)tricyclohexyltin(IV) have been prepared for the first time and have been found to be five-coordinate in the solid state. These and related five-coordinate complexes prepared previously have been studied by a variety of physical methods; ¹³C NMR, UV, IR, Raman, dipole moments and the Kerr effect. While all structures are demonstrably five-coordinate, and all chelates bidentate in the solid state, the geometries of two of the complexes in solution appear to vary somewhat from the expected fac or mer. There is evidence from the solution Kerr effect and ¹³C NMR that cyclohexyl derivatives may disproportionate.     

Pyrrolopyridazine nucleosides. The synthesis of certain 1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-ones

Nucleosides of pyrrolo[2,3-d]pyridazin-4(5H)-ones were prepared by the single-phase sodium salt glycosylation of appropriately functionalized pyrrole precursors. The glycosylation of the sodium salt of ethyl 4,5-dichloro-2-formyl-1H-pyrrole-3-carboxylate ( 4 ), or its azomethino derivative 7 , with 1-bromo-2,3,5-tri-O-benzoyl-D-ribofuranose in acetonitrile afforded the corresponding substituted pyrrole nucleosides ethyl 4,5-dichloro-2-formyl-1-(2,3,5-tri-O-benzoyl-β-D-ribofuranosyl)-1H-pyrrole-3-carboxylate ( 5 ) and ethyl 4,5-dichloro-2-phenylazomethino-1-(2,3,5-tri-O-benzoyl-β-D-ribofuranosyl)-1H-pyrrole-3-carboxylate ( 8 ), respectively. The latter, upon treatment with hydrazine, afforded the annulated product 2,3-dichloro-1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-one ( 6 ), in good yield. The unsubstituted analog 1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-one ( 9 ), was obtained upon catalytic dehalogenation of 6 . This report represents the first example of the synthesis of nucleosides of pyrrolopyridazines.     

Ring-closing metathesis of olefinic peptides: design, synthesis, and structural characterization of macrocyclic helical peptides

Heptapeptides containing residues with terminal olefin-derivatized side chains (3 and 4) have been treated with ruthenium alkylidene 1 and undergone facile ring-closing olefin metathesis (RCM) to give 21- and 23-membered macrocyclic peptides (5 and 6). The primary structures of peptides 3 and 4 were based upon a previously studied heptapeptide (2), which was shown to adopt a predominantly 3(10)-helical conformation in CDCl(3) solution and an alpha-helical conformation in the solid state. Circular dichroism, IR, and solution-phase (1)H NMR studies strongly suggested that acyclic precursors 3 and 4 and the fully saturated macrocyclic products 7 and 8 also adopted helical conformations in apolar organic solvents. Single-crystal X-ray diffraction of cyclic peptide 8 showed it to exist as a right-handed 3(10)-helix up to the fifth residue. Solution-phase NMR structures of both acyclic peptide 4 and cyclic peptide 8 in CD(2)Cl(2) indicated that the acyclic diene assumes a loosely 3(10)-helical conformation, which is considerably rigidified upon macrocyclization. The relative ease of introducing carbon-carbon bonds into peptide secondary structures by RCM and the predicted metabolic stability of these bonds renders olefin metathesis an exceptional methodology for the synthesis of rigidified peptide architectures.     

Solution properties of cellulose triacetate. II. Solubility and viscosity studies

The solubility of cellulose triacetate in a range of solvents was measured, and the results for tetrachloroethane, chloroform, and acetic acid were compared with those from initial phase separation in solvent–nonsolvent mixtures and viscosity–concentration studies. The correlation found between solubilities, precipitation values, and values of the Huggins viscosity constant is discussed with reference to the type of polymer–solvent interaction proposed previously to explain fractionation behavior. A qualitative comparison of solubility–swelling behaviour was also made for a very low molecular weight cellulose triacetate sample in a wide range of solvents. Results are compared with those for higher molecular weight samples and discussed with regard to the cohesiveenergy densities of solvent and polymer. Some attempt has been made to predict suitable solvents for cellulose triacetate, based on consideration of their molecular structures.     

Synthesis of 6,19-sulfamidate bridged pregnanes

Conformationally restrained substituted pregnane-20-one derivatives were obtained by an intramolecular nitrene addition onto a C-5/C-6 double bond involving a tethered C-19 sulfamoyl moiety. The resulting aziridine underwent regioselective nucleophilic ring opening at C-5 at room temperature with cyanide, fluoride, and acetate. In the isolated case of acetate, a reversal of regioselectivity was observed at higher temperatures, a result attributed to a rearrangement process involving aziridine ring opening at the C-5 position and subsequent migration of the acetyl moiety to C-6.