*Accepted Paper*

**Inserted:** 24 jul 2013

**Last Updated:** 18 may 2015

**Journal:** Rev. Mat. Iberoam.

**Volume:** 30

**Number:** 4

**Year:** 2014

**Doi:** 10.4171/rmi/817

**Abstract:**

We prove local Calderón-Zygmund estimates for weak solutions of the evolutionary $p(x,t)$-Laplacian system \[ \partial_t u-{\rm div}\,\big(a(x,t){\vert Du\vert}^{p(x,t)-2}Du\big) = {\rm div}\,\big(\vert F\vert^{p(x,t)-2}F\big) \] under the classical hypothesis of logarithmic continuity for the variable exponent $p(x,t)$. More precisely, we show that the spatial gradient $Du$ of the solution is as integrable as the right-hand side $F$, i.e. \[ \vert F\vert^{p(\cdot)}\in L^q_{\rm loc} \quad\Longrightarrow\quad \vert Du\vert^{p(\cdot)}\in L^q_{\rm loc} \qquad\text{for any $q>1$} \] together with quantitative estimates. Thereby, we allow the presence of eventually discontinuous coefficients $a(x,t)$, only requiring a VMO condition with respect to the spatial variable $x$.

**Keywords:**
gradient estimates, degenerate parabolic systems, non-standard growth condition

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