Asymptotic Behavior of a Non-Local Hyperbolic Equation Modelling Ohmic Heating

DOI：10.3770/j.issn:2095-2651.2012.04.012

 作者 单位 汪先超 国家数字交换系统工程技术研究中心, 河南 郑州 450002 江成顺 中南财经政法大学武汉学院, 湖北 武汉 430079

本文研究了基于欧姆加热模型的一类非局部双曲问题解的渐近性态.研究发现该双曲问题的解只有三种情况:问题解整体有界且其唯一稳态解渐近稳定;问题解无穷远爆破;问题解有限时刻爆破.如果问题解在有限时刻爆破,该解在(0,1]的任意子区间上一致爆破,且爆破速度为$\lim_{t\rightarrow T^{*}-}u(x,t)(T^{*}-t)^{\frac{1}{\alpha+\beta p-1}}=(\frac{\alpha+\beta p-1}{1-\alpha})^{\frac{1}{1-\alpha-\beta p}}$,这里$T^*$是爆破时间.

In this paper, the asymptotic behavior of a non-local hyperbolic problem modelling Ohmic heating is studied. It is found that the behavior of the solution of the hyperbolic problem only has three cases: the solution is globally bounded and the unique steady state is globally asymptotically stable; the solution is infinite when $t\rightarrow\infty$; the solution blows up. If the solution blows up, the blow-up is uniform on any compact subsets of $(0,1]$ and the blow-up rate is $\lim_{t\rightarrow T^{*}-}u(x,t)(T^{*}-t)^{\frac{1}{\alpha+\beta p-1}}=(\frac{\alpha+\beta p-1} {1-\alpha})^{\frac{1}{1-\alpha-\beta p}}$, where $T^{*}$ is the blow-up time.