Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case

We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-Δu+|Du|^γ=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $γ>2$, and $q_c=(d+2)(γ-1)/γ$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly