The authors present, in a unified way, various and interconnected basic aspects of the regularity theory of solutions to general nonlinear second order parabolic systems with polynomial \(p\)-growth, \(p\geq 2\), of the type \[ u_t-\text{ div}\, a(x,t,u,Du)=0 \] and related non-homogeneous ones \[ u_t-\text{ div}\, a(x,t,u,Du)+H=0,\quad H\in {\mathcal D'}(\Omega_T), \] under natural \(p\)-growth and ellipticity assumptions on the vector field \( a:\;\Omega_T\times{\mathbb R}^N\times{\mathbb R}^{Nn}\to {\mathbb R}^{Nn}\), that is \[ |a(x,t,u,w)|\leq L(1+|w|^2)^\frac{p-1}2\quad (x,t)\in \Omega_T, u\in\mathbb R^N, w\in \mathbb R^{Nn}, \] where \(\Omega_T=\Omega\times(-T,0)\), and \(\Omega\) is a bounded domain in \(\mathbb R^N.\) The authors give optimal partial regularity results for solutions and results on the Hausdorff dimension of the singular sets of solutions and the first Calderón-Zygmund-type results on higher integrability of solutions, also treating systems with possibly discontinuous coefficients.
New methods are introduced that allow to give the best possible forms of the rather preliminary versions of the desired regularity results already presented in the literature, on the one hand. On the other hand the authors hereby face several untouched issues: they give answers to a few non-clarified and difficult aspects of regularity theory, such as singular set reduction and general sharp gradient integrability estimates. Specifically: the partial regularity results in the interior are achieved via the method of \(A\)-caloric approximation, which extends DeGiorgi’s classical harmonic approximations lemma. Such a method for quadratic growth parabolic systems, i.e., \(p=2,\) is extended to cover general systems with polynomial super-quadratic growth \(p\geq 2\) and allows to deduce optimal partial regularity results for solutions in a direct way, without employing tools such as the reverse Hölder inequalities. The singular set estimates are obtained via a novel comparison method to deduce space-time fractional differentiability.
Finally, the gradient higher integrability results are new even for scalar parabolic equations (when considering scalar valued solutions \(N=1\)), or when applied to the basic model given by the classical evolutionary \(p\)-Laplacian system with (possibly discontinuous) coefficients \[ u_t-\text{div}\, (e(x,t)|Du|^{p-2}Du)=H\,. \] Moreover, in the case of general non-linear parabolic equations and of \(p\)-Laplacian-type systems, the optimal form of results available only for linear parabolic problems is found. Such results are obtained with harmonic-analysis-free proofs, since classical tools, such as singular integrals and maximal operators, cannot be used for evolutionary problems with polynomial growth. Nevertheless some intrinsic principles of harmonic analysis, such as local representation formulas and stopping time arguments, are employed directly at the suitable PDE level.