Bounded and periodic solutions of quasilinear parabolic equations in time-dependent domains

We show the existence and uniqueness of the bounded or periodic solution for the quasilinear parabolic equation of the form u(t)-div(sigma(|del u|(2))del u)=f(x,t) in Q(-infinity,infinity) (1.1) with the boundary condition u(t)|(partial derivative Omega)(t)=0, where Omega(t) is a bounded domain in R-N for each t is an element of R and Q(-infinity,infinity)=boolean OR-infinity<t= 0,sigma(v(2))=log(1+v(2)) and sigma(v(2))=|v|(m)/1+v(2),m >= 1. We derive a precise estimate for sup(-infinity<t2</SUP>+