Self-improving property of nonlinear higher order parabolic systems near the boundary

Abstract

We establish global regularity results for a wide class of non-linear higher order parabolic systems. The model problem we have in mind is the parabolic p-Laplacian system of order 2m, m ≥ 1, $$\partial_t u + (-1)^{m}\, {\rm div}^m \left(|D^mu|^{p-2}D^{m}u\right) = 0$$ with prescribed boundary and initial values. We prove that if the boundary values are sufficiently regular, then Dmu is globally integrable to a better power than the natural p. The method also produces a global estimate.