Very weak solutions of subquadratic parabolic systems with non-standard $p(x,t)$-growth

Abstract

The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic $p(x,t)$-Laplacian system. Under assumptions on the exponent function $p:{\Omega }_T=\Omega \times (0,T)\to \left(\frac{2n}{n+2},2\right]$, it is shown that any very weak solution $u:{\Omega }_T\to {\mathbb{R}}^N$ with $|Du{|}^{p(\cdot )(1-\epsilon )}\in L^1({\Omega }_T)$ belongs to the natural energy spaces, i.e. $|Du{|}^{p(\cdot )}\in L_{\mathrm{loc}}^1({\Omega }_T)$, provided $\epsilon >0$ is small enough. This extends the main result of [V. Bögelein and Q. Li, Nonlinear Anal., 98 (2014), pp. 190-225] to the subquadratic case.