Hölder estimate for the solution of degenerate parabolic equations

Consider the degenerate parabolic equations of the type $$u_t = div A(x,t,u,Du) + b(x,t,u,Du)$$ which is of the same nature as $$u_t = div|Du|^p Du + |Du|^{p + 2} (p > 2).$$ This paper is to study the \(C^{1 + \alpha } ,\frac{{1 + \alpha }}{2}\) Hölder continuity of a class of degenerate parabolic equations and the existence and uniqueness of the initial boundary value problem.     

Vibrational modes and infrared absorption of interstitial oxygen in silicon

A quasi-linear Si₂O molecule model (QLMM) is suggested from an analysis of the configuration and the interactions of an isolated oxygen atom with its neighbor silicon atoms. The vibrational modes are assigned and the infrared absorption spectra are calculated in detail with the model. The theoretical results are in reasonably good agreement with reported experimental values. This agreement shows that for the analysis of the vibrational modes of the interstitial oxygen atom in silicon crystals it is not necessary to consider the coupling of the molecule with the rest of the lattice. The interaction of the oxygen atom with its six second-nearest silicon atoms only causes the level separation of the ₂ mode and the formation of the fine structure.     

Alkylation of benzene with 1-hexene in acidic ionic liquid systems: Et3NHCl-FeCl3and Et3NHCl-AlCl3ionic liquids

Summary The alkylation of benzene with 1-hexene has been investigated in different triethylamine hydrochloride-ferric chloride (Et₃NHCl-FeCl₃) and triethylamine hydrochloride-aluminium chloride (Et₃NHCl-AlCl₃) ionic liquids. Both high catalyst activity and monoalkylation selectivity were observed for these two type of ionic liquids. Systems prepared by modification with HCl in Et₃NHCl-FeCl₃ionic liquids prove to be very suitable solvents and catalysts for the reaction. When employing Et₃NHCl-AlCl₃ionic liquids as catalysts, the reaction takes place in biphasic mode with facile catalyst separation and catalyst recycling.