Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation

The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|^p-2 uχ)χ + u |^q-2u(1-|u| ^r), defined in (0,1) X (0, +∞), is discussed. It is well known that there exists a stationary solution ø which has an open set Ω where it is identically ±1. We call a graph {(χ, ø (χ));χ ∈ Ω} flat hats. We investigate the behavior of u(χ,t) near (χ,t) ∈ Ω x [0,+∞) where |u(χ, t) - ø(χ)| is very small. We will give a sufficient condition for initial data U_o that the intersection points between the flat hats of ø and u never change as a function of t along u(',t;uo). Even if the condition failed, it is also proved that the changing area of the intersection points is uniformly bounded for t. Moreover we study stability properties for the positive stationary solution and the sign-changing stationary solutions.