Existence for general nonlinearities in the critical case

Let $f$ satisfy (F1) with $q\geq 1$, suppose that $f'(u)F(u)\leq q$ for all sufficiently large $u>0$, and let $\phi\geq 0$. Set

$$r=\frac{N}{2}>q-1.$$

If

$$F(\phi)^{-r}\in\mathcal{L}^1_{\mathrm{ul},\rho}(\mathbb{R}^N),$$

then (S1E1) has a local-in-time solution. Existence conjecture. The stated integrability condition should therefore suffice for local-in-time solvability for general nonlinearities in the critical case. This is left open in the paper; the preceding results establish corresponding existence statements for specific nonlinearities and under stronger assumptions.