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Gradient estimates for a class of parabolic systems. (English) Zbl 1113.35105
The very interesting paper under review deals with gradient estimates of Calderón–Zygmund type for a class of parabolic problems having as model the nonhomogeneous, degenerate/singular parabolic \(p\)-Laplacian system
\[ u_t-\text{div\,}(| Du| ^{p-2}Du)=\text{div\,}(| F| ^{p-2}F), \quad p>{{2n}\over{n+2}}, \tag{*} \]
considered in the cylinder \(C=\Omega\times[0,T)\) with bounded base \(\Omega\subset\mathbb R^n\) and
\(u\in C^0((0,T);L^2(\Omega,\mathbb R^N)) \cap L^p((0,T);W^{1,p}(\Omega,\mathbb R^N)),\) \(N\geq1,\) \(F\in L^p(\Omega,\mathbb R^{nN}).\) Adopting and intrinsic geometry viewpoint and certain Calderón-Zygmund covering arguments combined with \(C^{0,1}\)-estimates for the homogeneous \((F\equiv0)\) \(p\)-Laplacian system, the authors prove that
\[ F\in L^q_{\text{loc}} \Longrightarrow Du\in L^q_{\text{loc}}\quad \forall q\geq p \]
for any solution to \((*).\) The results obtained cover a more general class of systems of the type \[ u_t-\text{div\,}(a(x,t)| Du| ^{p-2}Du)=\text{div\,}(| F| ^{p-2}F),\quad a(x,t)\in \text{VMO/BMO}, \] and extensions involving operators different from the \(p\)-Laplacian are outlined as well.

MSC:
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35B45 A priori estimates in context of PDEs
35K40 Second-order parabolic systems
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