On the initial-boundary value problem for some quasilinear parabolic equations of divergence form

In this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form ut−div{σ(|∇u|2)∇u}=f(∇u,u,x,t) where σ(|∇u|2) may not be bounded as |∇u|→∞. As an application the logarithmic type nonlinearity σ(|∇u|2)=log⁡(1+|∇u|2) which is growing as |∇u|→∞ and degenerate at |∇u|=0 is considered under f≡0.