  We provide new conditions for the local existence of solutions to the
time-weighted parabolic equation $ u_t - \Delta u = h(t)f(u) \mbox{ in } \Omega
\times (0,T),$ where $ \Omega $ is a arbitrary smooth domain, $f\in
C(\mathbb{R})$, $h\in C([0,\infty))$ and $u(0)\in L^r(\Omega)$. As consequence
of our results, considering a suitable behavior of the non-negative initial
data, we obtain a second critical value $\rho^\star = 2r/(p-1),$ when
$f(u)=u^p$ and $p> 1 + 2r/N$, which determines the existence (or not) of a
local solution $u \in L^\infty((0,T), L^r(\Omega)).$
