Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

Abstract

The aim of this paper is to establish a Meyer's type higher integrability result for weak solutions of possibly degenerate parabolic systems of the type

$$ \partial_t u - \operatorname{div} a(x,t,Du)= \operatorname{div} \bigl(|F|^{p(x,t)-2}F\bigr).$$

The vector-field $a$ is assumed to fulfill a non-standard $p(x,t)$-growth condition. In particular it is shown that there exists $\varepsilon >0$ depending only on the structural data such that there holds:

$$|Du|^{p(\cdot)(1+\varepsilon)}\in L^1_{\operatorname{loc}},$$

together with a local estimate for the $p(\cdot)(1+\varepsilon)$-energy.