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 having a forcing term in Lebesgue spaces and sublinear gradient growth, 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 $L^m$, $m\neq 2$. This approach leads to new results even for stationary problems.