on a second critical value for the local existence of solutions in lebesgue spaces

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