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

We study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p(x,t)p(x,t)-Laplacian system, ∂tu−div(|Du|p(x,t)−2Du)=div(|F|p(x,t)−2F).∂tu−div(|Du|p(x,t)−2Du)=div(|F|p(x,t)−2F). Under natural assumptions on the exponent function p:Ω×(0,T)→[2,∞)p:Ω×(0,T)→[2,∞), we prove that any very weak solution u:Ω×(0,T)→RNu:Ω×(0,T)→RN with |Du|p(⋅)(1−ε)∈L1|Du|p(⋅)(1−ε)∈L1 belongs to the natural energy space, i.e. |Du|p(⋅)∈Lloc1, provided ε>0ε>0 is small enough.