Global higher regularity of solutions to singular p(x,t)-parabolic equations

We study the homogeneous Dirichlet problem for the equationut=div(|∇u|p(x,t)−2∇u)+f(x,t,u) in the cylinder QT=Ω×(0,T), Ω⊂Rd, d≥2. It is assumed that p(x,t)∈(2dd+2,2) and |∇p|, |pt| are bounded a.e. in QT. We find conditions on p(x,t), f(x,t,u) and u(x,0) sufficient for the existence of strong solutions, local or global in time. It is proven that the strong solutions possess the property of global higher regularity: ut∈L2(QT), |∇u|∈L∞(0,T;L2(Ω)), |Dij2u|p(x,t)∈L1(QT).