Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains

We consider the problem in Ω_ε, u=0 on ∂Ω_ε, where Ω_ε:=ΩB(0,ε) and Ω is a bounded smooth domain in , which contains the origin and is symmetric with respect to the origin, N3 and ε is a positive parameter. As ε goes to zero, we construct sign changing solutions with multiple blow up at the origin.     

A sharp form of Moser-Trudinger inequality in high dimension

Let Ω be a bounded smooth domain in Rn(n?3). This paper deals with a sharp form of Moser-Trudinger inequality. Let

     

Recent advances and applications of molecularly imprinted polymers in solid‐phase extraction for real sample analysis

Sample pretreatment is essential for the analysis of complicated real samples due to their complex matrices and low analyte concentrations. Among all sample pretreatment methods, solid‐phase extraction is arguably the most frequently used one. However, the majority of available solid‐phase extraction adsorbents suffer from limited selectivity. Molecularly imprinted polymers are a type of tailor‐made artificial antibodies and receptors with specific recognition sites for target molecules. Using molecularly imprinted polymers instead of conventional adsorbents can greatly improve the selectivity of solid‐phase extraction, and therefore molecularly imprinted polymer‐based solid‐phase extraction has been widely applied to separation, clean up and/or preconcentration of target analytes in various kinds of real samples. In this article, after a brief introduction, the recent developments and applications of molecularly imprinted polymer‐based solid‐phase extraction for determination of different analytes in complicated real samples during the 2015‐2020 are reviewed systematically, including the solid‐phase extraction modes, molecularly imprinted adsorbent types and their preparations, and the practical applications of solid‐phase extraction to various real samples (environmental, food, biological, and pharmaceutical samples). Finally, the challenges and opportunities of using molecularly imprinted polymer‐based solid‐phase extraction for real sample analysis are discussed.     

Blow‐up rate estimates for a parabolic system with multiple nonlinearities

In this paper, we study the blow‐up behaviors for the solutions of parabolic systems u_t=Δu+δ₁e, v_t=Δv+µ₁u in ?×(0, T) with nonlinear boundary conditions Here δ_i?0, µ_j?0, p_i?0, q_j?0 and at least one of δ_iµ_jp_iq_j>0(i, j=1, 2). We prove that the solutions will blow up in finite time for suitable ‘large’ initial values. The exact blow‐up rates are also obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Decay and nonexistence of global solutions of a quasilinear riser equation

We study qualitative properties of a quasilinear wave equation of fourth order that models the mechanical vibrations of a marine riser. Our analysis characterizes global and nonglobal solutions with respect to the norm of some Hilbert space, if energy is strictly less than the potential well depth. We employ invariant sets to show our results. In particular, we show that globality implies exponential decay to zero, and nonglobality is due to blow up. Both results are shown with respect to the norm of the solution in the Hilbert space. Copyright © 2009 John Wiley & Sons, Ltd.     

On a boundary controller of fractional type

In this paper we consider the evolution of sound in a compressible fluid with reflection of sound at the surface of the material. The boundary controller involves a derivative of non-integer order. It is proved that solutions blow up in finite time in the presence of a nonlinear source.     

标准算子代数上保因子交换性的映射

Let X, Y be real or complex Banach spaces with dimension greater than 2 and A, B be standard operator algebras on X and Y, respectively. Let φ ：A →B be a unital surjective map. In this paper, we characterize the map φ on .4 which satisfies （A - B）R = R（A-B） ξR （（A-B）→ （φ（B））φ（R） =φ（R）（（A）- （B）） for A, B, R E .4 and for some scalar     

A blow-up theorem for a discrete semilinear wave equation

In this paper, a discretization of a semilinear wave equation whose nonlinear term is a power function is investigated. It is shown that, when a condition on the initial value problem, similar to that governing the existence of blow-up solutions for the original continuous equation is met, the newly introduced difference equation has blow-up solutions with characteristics corresponding to those of the blow-up solutions for the original equation.