Blow-up rate for radially symmetric solutions of some parabolic equations with nonlinear boundary conditions

We consider the following parabolic equations with nonlinear boundary conditions: ut=Δu−a|u|p−1u in B1×(0,T), ∂νu=|u|q−1u on ∂B1×(0,T), where p,q>1, a⩾0 and B1 is the unit ball. We study the blow-up rate of radial symmetric solutions u(r,t) without any assumptions on the initial data. Furthermore we show the uniqueness of positive solutions of one dimensional backward self-similar solutions. As a consequence, we can determine the asymptotic behavior of blow-up solutions near the blow-up time.