### Abstract

The behavior of solutions u of the degenerate parabolic equation Ut = λ(|uχ|^{p-2} uχ)χ + u |^{q-2}u(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.

Original language | English |
---|---|

Pages (from-to) | 678-692 |

Number of pages | 15 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 31 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

### Keywords

- Comparison theorem
- Degenerate parabolic equation
- Flat hat
- Intersection comparison
- p-Laplace operator

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics