Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem $\left\{ ut=uxx+.(1.u).p,(x,t).(.1,1).(0,T),u(±1,t)=0,t.(0,T), u(x,0)=u0(x)<1,x.[.1,1] \right. $ either touches 1 in finite time or converges smoothly to a steady state as t . .. Some extensions of this result to higher dimensions are also discussed.