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In this paper, we consider a Stefan-like problem with a kinetic condition on the free boundary, i.e. a two-phase nonequilibrium problem in one dimension, and prove that it has a unique classical and global solution under some conditions. We also investigate the travelling wave solutions.

In this paper we deal with the one-dimensional Stefan problem ut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ;+, u(x, 0) =u0(x) with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if ∣f(u)∣Meγ∣u∣ for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u) Ceγ1∣u∣ for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if ∣f(u)∣Meγ∣u∣ for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u) Ce√2πe∣u∣.

Considering a solute transport problem described by some algebraic and partial differential equations with the presence of flux boundary conditions, we reduce the problem to a flxed point one and use a priori estimates to prove the existence and uniqueness of the global solutions.

The dielectric distributed around the surface of electrodes has a significant effect on the corona discharge. Corona discharge in power transmission lines in fine weather is different from that in rainy days as the surface condition is greatly changed. The surface of conductors is filled with the drops of rainwater; hence, the electric field around the conductors will be distorted. The use of a corona cage is an effective and economical means for duplicating corona progress in power transmission lines with lower voltage on the conductor in the cage. The practical conductor was utilized in this experiment under both fine and rainy conditions. The process of corona discharge was recorded and investigated. It was found that the corona discharge turned to be much more intense when rain drops were enlarged and more compressed around the conductor. The corona inception voltage, which is an important parameter of corona discharge, was also gained using an ultraviolet imager (UV imager). The corona inception voltage under the rainy condition was about 25% lower than that under the fine condition. The results of the comparison of audible noise and radio interference were analyzed.