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

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.