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In this paper we will apply the modified potential well method andvariational method to the study of the long time behaviors of solutions to aclass of parabolic equation of Kirchhoff type. Global existence and blow up infinite time of solutions will be obtained for arbitrary initial energy. To be alittle more precise, we will give a threshold result for the solutions to existglobally or to blow up in finite time when the initial energy is subcriticaland critical, respectively. The decay rate of the $L^2(\Omega)$ norm is alsoobtained for global solutions in these cases. Moreover, some sufficientconditions for the existence of global and blow-up solutions are also derivedwhen the initial energy is supercritical.

Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy