Monotone solutions to quasilinear parabolic equations

We study the behavior at large time of solutions to boundary value problems for quasilinear autonomous parabolic equations depending analytically on the unknown function and its deriva¬tives. Denote by $M$ the set of initial data such that if $u_0\in M$, then the solution $u(x,t;u_0)$ to the boundary value problem constructed for the initial data $u_0$ becomes strictly monotone for $t>\tau(u_0)$. It is proved that if the problem is dissipative, then the set $M$ contains an open dense subset. If the problem is not dissipative, then a necessary and sufficient condition for $M$ to be empty is obtained.