Threshold results of blow-up solutions to Kirchhoff equations with variable sources

The main goal of this work is to investigate an initial boundary value problem to a class of parabolic equation of Kirchhoff type with variable sources. Our objective is not only to provide a threshold result of blow-up in a finite time of solutions at a new subcritical energy but also to derive a new blow-up criterion. Additionally, we will calculate the lifespan and an upper bound estimation for blow-up time in different initial energy cases.