We consider a class of semilinear heat equations on $\mathbb R$,
including in particular the Fujita equation
\begin{equation*}
u_t=u_{xx} +|u|^{p-1}u,\quad x\in \mathbb{R},\ t\in\mathbb{R},
\end{equation*}
where $p>1$. We first give a simple proof and an extension of
a Liouville theorem concerning entire solutions with finite
zero number. Then we show that there is an infinite-dimensional
set of entire solutions with infinite zero number.