Stability of steady states for one dimensional parabolic equations with nonlinear boundary conditions

We consider one dimensional parabolic equations with nonlinear boundary conditions: u t = u x x − q u 2 q − 1 in R + × ( 0 , T ) , ∂ ν u = u q on { 0 } × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 in R + . This equation admits a family of positive stationary solutions { ϕ α ( x ) } α > 0 ( ϕ α ( 0 ) = α ) such that ϕ α 1 ( x ) < ϕ α 2 ( x ) if α 1 < α 2 . The main purpose of this paper is to study the stability of these stationary solutions. Furthermore we discuss the large time behavior of global solutions. In particular, we prove that every global solution is uniformly bounded and converges to one of the stationary solutions.