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:ΩT=Ω×(0,T)→(2nn+2,2], it is shown that any very weak solution u:ΩT→RN with |Du|p(⋅)(1−ε)∈L1(ΩT) belongs to the natural energy spaces, i.e. |Du|p(⋅)∈Lloc1(ΩT), provided ε>0 is small enough. This extends the main result of Bögelein and Li (2014) to the subquadratic case.