Let u be a solution of a second order semilinear parabolic equation which satisfies either Dirichlet or mixed boundary conditions. Let v be a stationary solution of this problem. Then by substituting $u = v + \varepsilon w$, and omitting terms of higher order in $\varepsilon $ than $O( \varepsilon )$, we may formally derive a linear problem. In this paper we discuss the asymptotic behavior of solutions u of the nonlinear problem in relation to such behavior of solutions w of the linear problem.We show that if, for some $\lambda _0 > 0$, $w = O( {e^{ - \lambda _0 t} } )$ as $t \to \infty $ (in the supremum norm), then $u - v = O( {e^{ - \lambda _0 t} } )$ as $t \to \infty $ for suitably restricted initial data, provided the nonlinear term in the original problem is sufficiently smooth in a neighborhood of v. An example is given which demonstrates that without this smoothness condition, the convergence of u towards v may take place at a slower rate than is the case for the linear problem.