Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions

This paper deals with the field enhancement due to insertion of a bow-tie structure of perfectly conducting inclusions into the two-dimensional space with a given field. The field enhancement is represented by the gradient blow-up of a solution to the conductivity problem. The bow-tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles. The domains are parts of cones near the vertices which are nearly touching to each other. We construct functions explicitly which characterize the field enhancement. As consequences, we derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners. The estimates show in quantitatively precise way that the field is enhanced beyond the corner singularities due to the interaction between two inclusions, and the blow-up rate is much higher than the one for the case of inclusions with smooth boundaries.     

带色散项的周期可积Hunter-Saxton方程的波的破裂

本文考虑一个带色散项的可积Hunter-Saxton方程的周期柯西问题.首先,我们得出该模型的强解的一个精确的爆破机制.其次,利用守恒量和特征方法,分别建立了有限时间内强解的爆破发生的充分条件.最后,给出了强解的爆破率.     

Singular Functions and Characterizations of Field Concentrations: a Survey

In the presence of closely located inclusions of the extreme material property, the physical fields, such as the electric field and the stress tensor, may be concentrated and arbitrarily large in the narrow region between two inclusions. Recently there has been significant progress on the quantitative characterization of the field concentration in the contexts of electrostatics(Laplace equation), linear elasticity(Lamé system), and viscous flow(Stokes system). This paper is to review such progress in a coherent way.     

Uniform blow-up rate for diffusion equations with localized nonlinear source

In this short paper, we investigate blow-up rate of solutions of reaction–diffusion equations with localized reactions. We prove that the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.     

非局部反应扩散方程组的爆破性质

本文讨论带Dirichlet边界条件的反应扩散方程组ut(x,t)=△u(x,t)+uα(x,t)．up(0,t),vt(x,t)=△v(x,t)+uβ(x,t)vq(0,t),研究了该问题正解的爆破性质并给出爆破集及其爆破速率．     

Note on the Blow-up Set for A Semilinear Parabolic Differential System

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Positive solutions of an elliptic partial differential equation on R

In this paper, we discuss the existence, nonexistence and uniqueness of positive solutions of a one-parameter family of elliptic partial differential equations on R^N (N>2). These equations are of interests in mathematical biology and Riemannian geometry. Our approach are based on variational arguments and comparison principles.     

一类高阶非线性波动方程解的存在性

研究一类高阶非线性波动方程的初边值问题 ,证明问题局部广义解的存在性、唯一性 ,并用凸性方法证明解爆破的充分条件 .     

Nondegeneracy and uniqueness for boundary blow-up elliptic problems

In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Δu=λa(x)u^p+g(x,u) in Ω, with u=+∞ on ∂Ω, has a unique positive solution for large enough λ, and determine its asymptotic behavior as λ→+∞. Here p>1, a(x) is a continuous function which can be singular near ∂Ω and g(x,u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u^p by e^u or a “logistic type” function f(u).     

A Discrete Phenomenon in the C and Analytic Regularity of the Blow-Up Curve of Solutions to the Liouville Equation in One Space Dimension

We give a complete discussion of the C^∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case.