Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space

In this paper, we consider a nonlinear divergence form parabolic equation with time-dependent coefficient and inhomogeneous Neumann boundary condition. We establish the new sufficient conditions on nonlinear functions to guarantee that the positive solution $u(xx,t)$ exists globally. Under the conditions to guarantee that the positive solution blows up, by establishing the Sobolev inequality in multidimensional space and constructing the unified functionals, we obtain upper and lower bounds of the blow-up time $t^{\ast }$.