Abstract

We prove Calder\'on-Zygmund estimates for a class of parabolic problems whose model is the non-homogeneous parabolic $p(x,t)$-Laplacian equation
\begin{align*}
\partial_t u-\text{div}\left(|Du|^{p(x,t)-2}Du\right)=f-\text{div}\left(|F|^{p(x,t)-2}F\right).
\end{align*}
More precisely, we will show that the spatial gradient $Du$ is as integrable as the inhomogeneities $f$ and $F$, i.e.
\begin{align*}
|F|^{p(x,t)},|f|^\frac{\gamma_1}{\gamma_1-1}\in L^q_\text{loc}~~~\Rightarrow~~~|F|^{p(x,t)}\in L^q_\text{loc}~~~\text{for any}~~~q>1,
\end{align*}
where $\gamma_1$ is the lower bound for $p(x,t)$. Moreover, it is possible to use this approach to establish the Calder\'on-Zygmund theory for parabolic obstacle problems with $p(x,t)$-growth.

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