Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

This paper considers a class of higher-order nonlocal parabolic equations with special coefficient. We apply the Faedo-Galerkin approximation method and cut-off technique to obtain the local solvability. Furthermore, based on the framework of the modified potential well, we get the global existence, asymptotic behavior, and blow-up of the weak solutions by the Hardy-Sobolev inequality when the initial energy is subcritical $J(u_0)<d$. In the critical case of $J(u_0)=d$, the above results have also been obtained. Finally, we utilize some new processing methods to gain the blow-up criterion in finite time with supercritical initial energy $J(u_0)>0$.