Global existence and blowup for sign-changing solutions of the nonlinear heat equation

In this paper, given 0<α<2/N, we prove the existence of a function ψ with the following properties. The solution of the equation ut−Δu=α|u|u on RN with the initial condition u(0)=ψ is global. On the other hand, the solution with the initial condition u(0)=λψ blows up in finite time if λ>0 is either sufficiently small or sufficiently large.     

Continuous Consecutive Reactions with Inter‐Reaction Solvent Exchange by Membrane Separation

Pharmaceutical production typically involves multiple reaction steps with separations between successive reactions. Two processes which complicate the transition from batch to continuous operation in multistep synthesis are solvent exchange (especially high‐boiling‐ to low‐boiling‐point solvent), and catalyst separation. Demonstrated here is membrane separation as an enabling platform for undertaking these processes during continuous operation. Two consecutive reactions are performed in different solvents, with catalyst separation and inter‐reaction solvent exchange achieved by continuous flow membrane units. A Heck coupling reaction is performed in N,N‐dimethylformamide (DMF) in a continuous membrane reactor which retains the catalyst. The Heck reaction product undergoes solvent exchange in a counter‐current membrane system where DMF is continuously replaced by ethanol. After exchange the product dissolved in ethanol passes through a column packed with an iron catalyst, and undergoes reduction (>99 % yield).     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

Sur la composition de l'arǒme de the noir V

Methyl anthranilate, geranylacetone and five lactones: γ-valerolactone, γ-heptalactone, γ-décalactone, δ-décalactone, 5-hydroxy dec-7-enoic acid-lactone have recently been identified in black tea aroma by means of GC./MS. and by sniffing the effluent from a gas chromatograph. Their intrinsic character, rather fruity and flowery, gives a certain freshness and an agreable note to the aroma.     

A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom

The system of ordinary differential equations

Optimized synthesis of tetrahydroisoquinoline-hydantoins

Several methods have been developed and compared for the solution synthesis of tetrahydroisoquinoline-hydantoin (Tic-hydantoin) derivatives. Starting materials were Tic-OH and amines readily available from commercial sources. The best yields were observed when the imidazolidine-2,4-dione ring was synthesized in two steps: (1) reaction of Tic-OH with the appropriate amine and (2) cyclization with 1,1′-carbonyldiimidazole.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,