The Singular Diffusion Parabolic Equation With Boundary Degeneracy

This thesis consists of three parts. The first part is the regularity of the solution for the singular diffusion parabolic equation with boundary degeneracy. The second part is the existence and uniqueness solution for the parabolic equation with absorption term. The last part is the large time behavior of solution for a degenerate parabolic equation.1. Consider the problem: where p>l,α>0. Then we proved that:(1) Estimate to the solution’s gradient bounded properties:(?), i=1,2.....N.(2) The Holder continuous of the solution: u(x1,t1)-u(x2,t2)|≤c(|x1-x2|+|t1-t21/2|where c is a constant only dependent on N, p,d(x,(?)Ω) and||u||L∞(K), K(?)QTT is compact set.(3) The solution’s gradient locally Holder continuous in QT.2. Degenerate parabolic equations with absorption term: where p>1, q>0. Then we obtain that:(1)If a≥p-1, the equation admits a unique solution subject only to a given initial datum without any boundary value condition.(2)If0<α<p-1, for a given initial datum, the equation admits different solutions for different boundary value conditions.3. When p>2, then consider the large time behavior of solution for the following degenerate parabolic equation: Conclusion1:u(x,t)≥C(t+1)-r,when(x,t)∈Ω×(0,∞),r=(p-2)-1. Conclusion2:u(x,t)≥Cφ(x)(t+1),when(x,t)∈Ω×(0,∞),r=(p一2)-1where C is a constant only dependent ond(x,(?)Ω),p andφ.