  A reaction-diffusion equation is studied in a time-dependent interval whose
length varies with time. The reaction term is either linear or of KPP type. On
a fixed interval, it is well-known that if the length is less than a certain
critical value then the solution tends to zero. When the domain length may vary
with time, we prove conditions under which the solution does and does not
converge to zero in long time. We show that, even with the length always
strictly less than the 'critical length', either outcome may occur. Examples
are given. The proof is based on upper and lower estimates for the solution,
which are derived in this paper for a general time-dependent interval.
