In this paper, we study the solution of an initial boundary value
problem for a quasilinear parabolic equation with a nonlinear
boundary condition. We first show that any positive solution blows
up in finite time. For a monotone solution, we have either the
single blow-up point on the boundary or blow-up on the whole domain,
depending on the parameter range. Then, in the single blow-up point
case, the existence of a unique self-similar profile is proven.
Moreover, by constructing a Lyapunov function, we prove the
convergence of the solution to the unique self-similar solution as
$t$ approaches the blow-up time.