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

Abstract

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 $.

Keywords

• logarithmic nonlocal source
• special diffusion process
• global existence
• asymptotic behavior
• blow-up