We study the large time behavior of positive solutions of the semilinear parabolic equation $u_t = u_{xx} + \varepsilon (g(u))_x + f(u)$, $0 < x < L$, $\varepsilon \in {\bf R}$, subject to $u(0,t) = u(L,t) = 0$. The model problem in which the results apply is $g(u) = u^m $ and $f(u) = u^p 1 \leqq m < p$. The steady state problem is analyzed in some detail, and results about finite time blow up are proved.