The inequality
(
∂
t
u
−
Δ
u
)
(
t
,
x
)
≤
u
(
t
,
x
)
(
1
−
u
(
t
−
τ
,
x
)
)
({\partial _t}u - \Delta u)(t,\,x)\qquad \leq \qquad u(t,\,x)(1 - u(t - \tau ,\,x))
is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.