Quasilinear parabolic equations with nonlinear boundary conditions

Of concern is the following quasilinear parabolic equation with nonlinear boundary conditions: (∗)ut(x,t)=α(x,Du)Δu+g(x,u,Du) for (x,t)∈Ω×(0,∞)∂u∂ν+β(x,u)=0 for (x,t)∈∂Ω×(0,∞)u(x,0)=u0(x).The diffusion coefficient α can vanish on the spacial boundary at a certain rate. It is shown by a difference scheme from the method of lines that (∗) has a unique global strong solution.