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

Abstract

We provide new conditions for the local existence of solutions to the time-weighted parabolic equation $ u_t - Δu = h(t)f(u) \mbox{ in } Ω\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 $u\in L^{\infty }((0,T),L^r(\Omega )).$