  We are concerned with solutions to the nonlinear heat equation $u_t=\Delta
u+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and
negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren
exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady
state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we
show that $u$ is independent of time. Our method of proof uses Serrin's
sweeping principle, based on the strong maximum principle, applied to the
linearized equation for $u_t$. Our result covers that of Pol\'{a}\v{c}ik and
Yanagida [JDE (2005)] who had further assumed that the solution stays above
some positive radial steady state and $p>p_c$. In contrast, they relied on the
use of similarity variables and invariant manifold ideas. Remarkably, to the
best of our knowledge, a corresponding Liouville property was previously
missing for $p =p_c$. We emphasize that such Liouville type theorems imply the
quasiconvergence of a class of solutions to the corresponding Cauchy problem.
As our viewpoint originates from the study of elliptic problems, we can prove
new rigidity results for the corresponding steady state problem that are
motivated by the aforementioned ones for the parabolic flow.
