We consider the homogeneous Dirichlet problem for anisotropic parabolic equationut−∑i=1NDxi(|Dxiu|pi(x,t)−2Dxiu)=f(x,t) in cylinder Ω×(0,T), where Ω⊂RN, N≥2, is a parallelepiped. The exponents of nonlinearity pi are given Lipschitz-continuous functions. It shown that if pi(x,t)>2NN+2,μ=supQTmaxipi(x,t)minipi(x,t)<1+1N,|Dxiu0|max{pi(⋅,0),2}∈L1(Ω),f∈L2(0,T;W01,2(Ω)), then has unique solution ...