The asymptotic self-similar behavior for the quasilinear heat equation with nonlinear boundary condition

In this paper the quasilinear heat equation with the nonlinear boundary condition is studied. The blow-up rate and existence of a self-similar solution are obtained. It is proved that the rescaled function v(y,t)=(T−t)1/(2p+α−2)u((T−t)(p−1)/(2p+α−2)y,t),v(y,t)=(T−t)1/(2p+α−2)u((T−t)(p−1)/(2p+α−2)y,t), behaves as t→Tt→T like a nontrivial self-similar profile.