Global existence and blowup for quasilinear parabolic equations not in divergence form

This paper deals with global and blowup solutions of the quasilinear parabolic equation u t = α ( x , u , ∇ u ) Δ u + f ( x , u , ∇ u ) with homogeneous Dirichlet boundary conditions. We will give sufficient conditions such that the solutions either exist globally or blow up in a finite time for any smooth initial values. In special cases, a necessary and sufficient condition for global existence is given.