Asymptotic Behavior of a Non-Local Hyperbolic Equation Modelling Ohmic Heating
Received:December 28, 2010  Revised:December 19, 2011
Key Words: non-local hyperbolic equation   asymptotical behavior   blow-up   blow-up rate.  
Fund Project:Supported by the National High Technology Research and Development Program of China (Grant No.2012AA011603).
Author NameAffiliation
Xianchao WANG National Digital Switching System Engineering & Technological R & D Centre, Henan 450002, P. R. China 
Chengshun JIANG Wuhan College, Zhongnan University of Economics and Law, Hubei 430079, P. R. China 
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Abstract:
      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.
Citation:
DOI:10.3770/j.issn:2095-2651.2012.04.012
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