| In this dissertation, we study interfaces of the parabolic p-Laplacian equation {dollar}usb{lcub}t{rcub}=Deltasb{lcub}p{rcub}u{dollar} for {dollar}p>2.{dollar} We establish their {dollar}Csp{lcub}1,alpha{rcub}{dollar} regularity after a large time under some suitable conditions on the initial date. It is well known that the solution u and {dollar}nabla uin Csp{lcub}alpha{rcub}{dollar} for all {dollar}t>0, xin Rsp{lcub}N{rcub}.{dollar} Hence {dollar}nabla u=0{dollar} on the interface, which unfortunately does not give information on the interface itself. The key idea is to study a new function {dollar}v={lcub}{lcub}p-2{rcub}over{lcub}p-1{rcub}{rcub}usp{lcub}{lcub}p-2{rcub}over{lcub}p-1{rcub}{rcub}{dollar} which has the same interface as u. The lower and upper bounds for {dollar}vsb{lcub}t{rcub}{dollar} and {dollar}vertnabla vvert{dollar} can be estimated near the interface by constructing some lower and upper solutions. Consequently some standard theory on strongly parabolic equations can be used for v. In addition, some useful and interesting properties of v are also obtained along the way to prove the main theorem. |
