Extinction And Positivity For P-Laplacian Parabolic Equation And Global Existence For P-Laplacian Parabolic System

In this paper, we study the extinction and positivity for fast diffusion p-Laplacian equations with nonlinear sources and global existence for p-Laplacian parabolic system. The paper contains two chapters.In chapter I, we study behaviors of solution for fast diffusion p- Laplacian equationwhere 1 < p < 2, λ > 0, m > 0. Our main results are as follows: (1) If p - 1 < m ≤ 1, then the solution with small initial data will vanish in finite time;(2) If p - 1 > m, then the maximal weak solution U(x, t) with nonnegative nontrival initial data cannot vanish in finite time;(3) if p - 1= m, then the minimal positive eigenvale λ₁ of ellipiptic p-Laplacian equation: -div in Ω, ψ(x) = 0 on (?)Ω. plays a critial role;(4) If m > 1, the solution with large initial data blows up in finte time. In chapter II, we prove that the solution of the considered p-Laplacian parabolic system extists for all T > 0.