Paper detail

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

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 $p:Ω_T=Ω\times (0,T)\to\left(\frac{2n}{n+2},2\right]$, it is shown that any very weak solution $u:Ω_T\rightarrow\mathbb{R}^N$ with $|Du|^{p(\cdot)(1-\varepsilon)}\in L^1(Ω_T)$ belongs to the natural energy spaces, i.e. $|Du|^{p(\cdot)}\in L^1_{\operatorname{loc}}(Ω_T)$, provided $ε>0$ 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.