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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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