Uniqueness of the renormalized solutions to the Cauchy problem for an anisotropic parabolic equation

We consider the Cauchy problem for a certain class of anisotropic parabolic second-order equations with double non-power nonlinearities. The equation contains an “inhomogeneity” in the form of a non-divergent term depending on the sought function and spatial variables. Non-linearities are characterized by $N$-functions, for which $Delta_2$-condition is not imposed. The uniqueness of renormalized solutions in Sobolev–Orlich spases is proved by the S. N. Kruzhkov method of doubling the variables.