On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas

We measure the zero-temperature equation of state of a homogeneous Bose gas of (7)Li atoms by analyzing the in situ density distributions of trapped samples. For increasing repulsive interactions our data show a clear departure from mean-field theory and provide a quantitative test of the many-body corrections first predicted in 1957 by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957).]. We further probe the dynamic response of the Bose gas to a varying interaction strength and compare it to simple theoretical models. We deduce a lower bound for the value of the universal constant ξ > 0.44(8) that would characterize the universal Bose gas at the unitary limit.     

Formation of an in situ nanosilica nanomatrix via graft copolymerization of vinyltriethoxysilane onto natural rubber

Natural rubber (NR) with an in situ nanosilica nanomatrix was characterized in present work. The in situ nanosilica nanomatrix was prepared via graft copolymerization of a silane monomer, vinyltriethoxysilane (VTES), onto deproteinized NR (DPNR) in latex stage using tetrapentamine (TEPA)/tert‐butylhydroperoxide (TBHPO) as initiators. VTES conversion of more than 80% was obtained, and it depended on VTES concentration. The graft copolymer structure was characterized by Fourier transform infrared (FT‐IR), solution‐state proton nuclear magnetic resonance (¹H‐NMR), and solid‐state ²⁹Si‐NMR spectroscopy. FT‐IR analysis of the graft copolymer confirmed the formation of in situ silica particles, while solution‐state ¹H‐NMR and solid‐state ²⁹Si‐NMR revealed the partial hydrolysis of the ethoxy groups and polycondensation of the silanol groups. The formation of nanosilica particles enhanced thermal and mechanical properties of the graft copolymer. Morphology observations of the in situ nanosilica nanomatrix through scanning electron microscopy (SEM) and transmission electron microscopy (TEM) indicated that the spherical nanosilica particles form a nanomatrix surrounding NR particle. The formation of the nanomatrix was proved to enhance mechanical properties for NR materials.     

Preparation of polyurethane dispersions by miniemulsion polymerization

With a two‐step miniemulsion polymerization, hydrophobic polyurethane (PU) dispersions were prepared with a cosurfactant, the costabilizer hexadecane (HD) in the oil phase, and sodium dodecyl sulfate (SDS) in the water phase. The first step involved the formation of NCO‐terminated prepolymers between isophorone diisocyanate and poly(propylene glycol) oligomer in toluene. Next, PU dispersions were produced by a miniemulsion method in which an oil phase containing NCO‐terminated prepolymers, HD, the chain extender 1,4‐butanediol (BD), the crosslinking agent trimethylol propane (TMP), and the catalyst dibutyltin dilaurate was dispersed in the water phase containing SDS. The influence of experimental parameters, such as the ultrasonication time, concentrations of SDS and HD, and TMP/BD and NCO/OH equivalent ratios, on the sizes of the miniemulsion droplets and polymer particles, as well as the molecular weights and thermal properties of the PU polymer, was examined. The chemical structure of the produced PU polymer was identified with a Fourier transform infrared spectrometer. The molecular weight distribution and average particle size were measured through gel permeation chromatography and dynamic light scattering, respectively. The thermal stability of the PU polymer was characterized with thermogravimetric analysis. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 4870–4881, 2005     

A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates

In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

A simple regularization method for the ill-posed evolution equation

The nonhomogeneous backward Cauchy problem

Dynamics of species in a model with two predators and one prey

In this paper, we study a predator-prey model which has one prey and two predators with Beddington-DeAngelis functional responses. Firstly, we establish a set of sufficient conditions for the permanence and extinction of species. Secondly, the periodicity of positive solutions is studied. Thirdly, by using Liapunov functions and the continuation theorem in coincidence degree theory, we show the global asymptotic stability of such solutions. Finally, we give some numerical examples to illustrate the behavior of the model.