汪先超,江成顺.基于欧姆加热模型的一类非局部双曲方程解的渐近性态[J].数学研究及应用,2012,32(4):476~484
基于欧姆加热模型的一类非局部双曲方程解的渐近性态
Asymptotic Behavior of a Non-Local Hyperbolic Equation Modelling Ohmic Heating
投稿时间:2010-12-28  最后修改时间:2011-12-19
DOI:10.3770/j.issn:2095-2651.2012.04.012
中文关键词:  非局部双曲方程  渐近性态  爆破  爆破速率.
英文关键词:non-local hyperbolic equation  asymptotical behavior  blow-up  blow-up rate.
基金项目:国家高技术研究发展计划``863计划'' (Grant No.2012AA011603).
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 
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中文摘要:
      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.
英文摘要:
      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.
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