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 equationAs consequence of our results, considering a suitable behavior of the non-negative initial data, we obtain a second critical value ρ ⋆ = 2r/(p − 1), when f (u) = u p and p > 1 + 2r/N , which determines the existence (or not) of a local solution u ∈ L ∞ ((0, T ), L r (Ω)).