Asymptotic behavior of solutions of second order parabolic partial differential equations with unbounded coefficients

Classical solutions of a second order parabolic partial differential equation are considered in unbounded domains. The coefficients are allowed to have unbounded growth as the space variables tend to infinity. A Phragmèn-Lindelöf principle is proved for such equations. That principle is used together with comparison functions to derive sufficient conditions for the asymptotic decay in time of solutions.     

Selective oxidation of benzylic and allylic alcohols using Mn(OAc)3/catalytic 2,3-dichloro-5,6-dicyano-1,4-benzoquinone

A practical, chemoselective oxidation of alcohols employing catalytic quantities of DDQ as the oxidant and Mn(OAc)(3) as the co-oxidant is described. Electron-rich benzylic alcohols are oxidized efficiently to their corresponding carbonyls, but less electron-rich benzylic alcohols remain unchanged. Allylic alcohols are rapidly oxidized to their corresponding aldehyde or ketone counterparts in high yields. This protocol is operationally simple, employs an inexpensive source of Mn(OAc)(3), has short reaction times, and exhibits a significant chemoselectivity favoring allylic alcohols over benzylic alcohols. This procedure also avoids the use of very large excesses of reagents and sometimes poor reproducibility that characterize previously developed reagents such as MnO(2).     

Populations with individual variation in dispersal in heterogeneous environments: Dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic-type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reactiondiffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that "the slower diffuser wins".     

Approximating the ideal free distribution via reaction-diffusion-advection equations

We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions.     

Concise, convergent syntheses of (±)-trichostatin A utilizing a Pd-catalyzed ketone enolate α-alkenylation reaction

Two concise, convergent syntheses of (±)-trichostatin A (1), a potent histone deacetylase inhibitor, have been accomplished. The key step in both is a Pd-catalyzed α-alkenylation reaction between ketone 2 and either dienyl bromide 3 or alkenyl bromide 9 using a modification of cross-coupling conditions described by Negishi and Hartwig. A brief investigation has shown the potential utility of a Ni-catalyzed version of this reaction. The overall synthetic routes are short and amenable to scaleup, providing access to trichostatin A via trichostatic acid as a direct precursor.     

Does movement toward better environments always benefit a population?

We study the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.     

Some estimates of the norm of solutions of nonlinear elliptic eigenvalue problems

We obtain estimates on the possible growth or decay rates as λ → 0 of sup |u^λ|, where uλ ? O satisfies the nonlinear elliptic boundary value problen Lu_λ = λ f(x,u_λ) in a bounded domain subject to homogensous Dirichlet boundary conditions. The estimates generalize existing results by allowing f(x,O) ≠ 0. The analysis is based on integration by parts and Sobolev inequalitie.