On maximal regularity estimates for quasilinear evolution equations via the integral Bernstein method

This work addresses the problem of (global) maximal regularity for quasilinear evolution equations with sublinear gradient growth and right-hand side in Lebesgue spaces, complemented with Neumann boundary conditions. The proof relies on a suitable variation of the Bernstein technique and the Bochner identity, and provides new results even for the simpler parabolic $p$-Laplacian equation with unbounded source term. As a byproduct we also obtain a second-order estimate that can be of independent interest when the right-side of the equation belongs to $L^m$, $m\mathrm{\ne }2$. This approach leads to new results even for stationary problems.