Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy \begin{document}$ J(u_0)\leq d $\end{document} . When the initial energy \begin{document}$ J(u_0)>d $\end{document} , we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.