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

Abstract

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 ∆ 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.

Discrete solution for the nonlinear parabolic equations with diffusion terms in Museilak-spaces

“…Recently, the uniqueness of renormalized solution of (1) in the general case has been proven by A. Aberqi et al in [9] and by F. Kh. Mukminov in [10,11] for the Cauchy problem for anisotropic parabolic equation using Kruzhkov's method of doubling the variable.…”

Descret Solution for a Nonlinear Parabolic Equations with Diffusion Terms in Museilak-Spaces

“…Aberqi et al in [1] and by F. KH. Mukminov in [20,21] for the Cauchy problem for anisotropic parabolic equation using Kruzhkovis method of doubling the variable. Concerning the Musielak spaces, these are spaces that generalize the Orlicz spaces, the Lebesgue spaces with weight, and the Lebesgue spaces with variable exponent, we refer to [22].…”