We consider parabolic equations of the form 

$\quad
u_t- \Delta u=h(x_1)f(u), \quad x=(x_1,x_2,\dots,x_N)\in R^N,\ t\in R,
$

where $h$ and $f$ are nondecreasing continuous functions,
$h(0)=0$,  $h$ is strictly increasing and unbounded for $x_1>0$. 
Our main goal is to prove a Liouville type result to the effect that
the above problem has no positive bounded entire solutions.
As an application of this result, we improve an earlier theorem
on complete blow-up for indefinite superlinear parabolic equations.