Very weak solutions of higher-order degenerate parabolic systems

Abstract

We consider non-linear higher-order parabolic systems whose simplest model is the parabolic $p$-Laplacean system \begin{equation*} \int_{\Omega_T} u\cdot \varphi_t - \langle |D^mu|^{p-2}D^mu,D^m\varphi\rangle \,dz = 0. \end{equation*} It turns out that the usual regularity assumptions on solutions can be weakened in the sense that going slightly below the natural integrability exponent still yields a classical weak solution. Namely, we prove the existence of some $\beta>0$ such that $D^mu\in L^{p-\beta} \Rightarrow D^mu\in L^{p+\beta}$.