We study the homogeneous Dirichlet problem for the equation ut=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)∈([Formula presented],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)), |Dij 2u|p(x,t)∈L1(QT).