The global weak solution for the shallow water wave model of moderate amplitude

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.     

Facile preparation of a polydopamine‐based monolith for multiple monolithic fiber solid‐phase microextraction of triazine herbicides in environmental water samples

A new multiple monolithic fiber solid‐phase microextraction using a polydopamine‐based monolith as the extraction medium is proposed. The monolith was synthesized by facile in situ copolymerization of N‐methacryldopamine and dual cross‐linkers (divinylbenzene/ethylenedimethacrylate) in the presence of N ,N‐dimethylformamide. The effect of the contents of N‐methacryldopamine and porogen in the polymerization mixture on the extraction performance was investigated thoroughly. A series of characterization studies was performed to validate the structure and properties of the monolith. The prepared multiple monolithic fibers were used for the extraction of triazine herbicides in environmental water samples. After the optimization of the extraction parameters, a convenient, sensitive, cost‐effective, and environmentally friendly method for the determination of trace triazine herbicides in water samples was developed by coupling multiple monolithic fibers solid‐phase microextraction with high‐performance liquid chromatography and diode array detection. The results indicated that the limits of detection and quantification for the target compounds were 0.031–0.14 and 0.10–0.45 μg/L, respectively. Good precision and reproducibility were obtained with the relative standard deviations below 10%. The developed method was applied to the analysis of the triazine herbicides in different water samples (lake, river, and farmland waters). The recoveries of the method were in the range between 79.6 and 117%.     

Micro‐ and Nano‐Liquid Phases Coexistent with Ice as Separation and Reaction Media.

Ice has a variety of scientifically interesting features, some of which have not been reasonably interpreted despite substantial efforts by researchers. Most chemical studies of ice have focused on the elucidation of its physicochemical nature and its roles in the natural environment. Ice often contains impurities, such as salts, and in such cases, a liquid phase coexists with solid ice over a wide temperature range. This impure ice also acts as a cryoreactor, governing the circulation of chemical species of environmental importance. Reactions and phenomena occurring in this liquid phase show features different from those seen in normal bulk aqueous solutions. In the present account, we discuss the chemical characteristics of the liquid phase that develops in a frozen aqueous phase and show how novel analytical systems can be designed based on he features of the liquid phase which are predictable in some cases but unpredictable in others.     

Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.     

带有Sobolev-Hardy临界指标项的非齐次椭圆方程的解

考虑带有Hardy和Sobolev-Hardy临界指标项的非齐次椭圆方程{-Δu-u(u/(|x|~2))=λu+(((|u|~(2~*(s)-2))/(|x|~s))u+f,在Ω中,u=0,在Ω上,这里2~*(s)=(2(N-s))/(N-2)是临界Sobolev-Hardy指标,N≥3,0≤s2,0≤μ=((N-2)~2)/4,ΩR~N是一个开区域.假设0≤λ≤λ_1时,λ_1是正算子-△-μ/(|x|~2)的第一特征值.f∈H~1_0(Ω)~*,f(x)≠0.当f满足适当的条件时,此方程在H~1_0(Ω)中至少具有两个解u_0和u_1.而且,当f≥0时,有u_0≥0和u_1≥0.     

Existence of multiplicity harmonic and subharmonic solutions for second‐order quasilinear equation via Poincaré‐Birkhoff twist theorem

In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second‐order quasilinear equation where , g satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase‐plane analysis methods and a generalized version of the Poincaré‐Birkhoff twist theorem.     

Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a (weakly) LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a (weakly) LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval-objective function are convex.